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THE 



PRACTICAL 

MODEL CALCULATOR, 



ENaiNEER, MECHANIC, MACHINIST, 

MANUFACTURER OF ENaiNE-WORK, NAVAL ARCHITECT, 

MINER, AND MILLWRIGHT. 



OLIVER BYRNE, 

CIVIL, MILITARY, AND MECHANICAL ENGINEER, 

Conipilf7- mid Editor of the "Dictionary of Machines, Mechanics, Engine-^vork, and Engineering ; 

Author of " The Companion for Machinists, Mechanics, and Engineers ;'" Avthor and Inventor 

of a Neio Science., termed "■The Calculus of Form," a substitute for tlia differential 

and Integral Calculus; " The Elements of Eudid hy Colours,'" and numerous 

other Mathematical and Mechanical Worlcs. Surveyor-General of the 

English Settlements in the Falklcnid Isles. Professor of 

Mathematics, College of Civil Engineers, London. 



PHILADELPHIA: 

HENRY CAREY B A I R D, 

40G WALNUT STRKKT. 

18G8. 






\8e3 



Entered according to the act of Congress, in the year 1851, by 
HENRY CAREY BAIRD, 

in the Clerk's Office of the District Court for the Eastern District of Penns3'lTania. 



THE 

PRACTICAL MODEL CALCULATOR. 



WEIGHTS AND MEASURES. 



THE UNIT OF LENGTH. 

The Yard. — If a pendulum vibrating seconds in vacuo, in Phi- 
ladelphia, be divided into 2509 equal parts, 2310 of such equal 
parts is the length of the standard yard ; the measures are taken 
on brass rods at the temperature of 32° Fahrenheit. This yard 
will not be in error the ten-millionth part of an inch. 
2310 : 2509 as 1- to 1-086142 nearly. 

THE UNIT OF WEIGHT. 

The Pound, avoirdupois, is 27*7015 cubic inches of distilled 
water, weighed in air, at the temperature of maximum density, 
39°-82 ; the barometer at 30 inches. 

THE LIQUID UNIT. 

The G-allon, 231 cubic inches, contains 8*3388822 pounds avoir- 
dupois, equal 58372*1754 grains troy of distilled water, at 39°-82 
Fah. ; the barometer at 30 inches. 

UNIT OF DRY CAPACITY. 

The Bushel contains 2150*42 cubic inches, 77*627412 pounds 
avoirdupois, 543391*89 grains of distilled water, at the temperature 
of maximum density ; the barometer at 30 inches. 

The French unit of length or distance is the metre, and is the 
ten-millionth of the quadrant of the globe, measured from the 
equator to the pole. 

The French Metre = 3*2808992 English /ee^ linear measure = 
39*3707904 inches. 

For Divisors the following Latin 
words are used : 

Bed for the lOi^^ part. 

Centi — lOO^A part. 

Milli — lOOOi/i part. 

Thus a Kilometre = 1000 metres. 
^_„. metre 

Imllimetre = TaaTT 

The square Beca Metre, called the Are, is the element of land 
measure in France, which = 1076*42996 square feet English. 
The Stere is a cubic metre = 35*316582 cubic feet English. 
a2 6 



For Multiples the following G-reek 
words are used : 

Beca for 10 times. 

llecto — 100 times. 

Kilo — 1000 times. 

Myria— lOOOOtimes. 



THE PRACTICAL MODEL CALCULATOR. 

The Litre for liquid measure is a cubic decimetre = 1-76077 
rial pints English, at the temperature of melting ice ; a litre 
of distilled water weighs 15434 grains troy. 

The unit of weight is the gramme : it is the weight of a cubic 
centimetre of distilled water, or of a millilitre, and therefore equal 
to 15*434 grains troy. 

The kilogramme is the weiorht of a cubic decimetre of distilled 
water, at the temperature of maximum density, 4° centigrade. 

The pound troy contains 5760 grains. 

The pound avoirdupois contains 7000 grains. 

The English imperial gallon contains 277*274 cubic inches ; and 
the English corn bushel contains eight such gallons, or 2218'192 
cubic inches. 

APOTHECAHIES' WEIGHT. 

Grains marked gr. 

20 Grains make 1 Scruple — sc. or 9 

3 Scruples — 1 Dram — dr. or 5 

8 Drams — 1 Ounce — oz. or g 

12 Ounces — 1 Pound — lb. or ft). 

gr. sc. 

20 = 1 dr. 

60 = 3=1 oz. 

480 = 24 = 8 = 1 lb. 
5760 = 288 = 96 = 12 = 1 
This is the same as troy weight, only having some different 
divisions. Apothecaries make use of this weight in compounding 
their medicines ; but they buy and sell their drugs by avoirdupois 
weight. 

AVOIRDUPOIS WEIGHT. 



Drams 




.marked dr. 


16 Drams 


.... make 1 Ounce 


. — oz. 


16 Ounces 


.... — 1 Pound 


. — lb. 


28 Pounds 


.... — 1 Quarter 


. — or. 


4 Quarters 


1 TTiiTirlrPfl WpiVlit . 


— cwt. 


20 Hundred Weight... — 1 Tnn °.... 


— ton. 


dr. 
16 = 

256 = 

7168 = 

28672 = 

573440 = 


oz. 

1 lb. 
16 = 1 qr. 
448 = 28 = 1 cwt. 
1792 = 112 = 4 = 1 
35840 = 2240 = 80 = 20 = 


ton. 
= 1 



By this weight are weighed all things of a coarse or drossy 
nature, as Corn, Bread, Butter, Cheese, Flesh, Grocery "Wares, 
and some Liquids ; also all Metals except Silver and Gold. 

Oz. Dwt. Gr. 
Note^ that 1 lb. avoirdupois = 14 11 15J troy. 
1 oz. — = 18 51 — 

Idr. — = 1 31 — 



WEIGHTS AND MEASURES. 



TEOY WEIGHT. 



Gr. Dwt. 
24 = 1 Oz. 
480 = 20 = 1 Lb. 
5760 = 240 = 12 = 1 



Grains marked Gr. 

24 Grains make 1 Pennyweight Dwt. 
20 Pennyweights 1 Ounce Oz. 

12 Ounces 1 Pound Lb. 

By this weight are weighed Gold, Silver, and Jewels. 

LONG MEASURE. 

3 Barley-corns make 1 Inch marked In. 

12 Inches — 1 Poot — Ft. 

3 Feet — 1 Yard — Yd. 

6 Feet — 1 Fathom — Fth. 

5 Yards and a half — 1 Pole or Rod — PI. 

40 Poles — 1 Furlong — Fur. 

8 Furlongs — 1 Mile — Mile. 

3 Miles — 1 League — Lea. 

69 1 Miles nearly — 1 Degree — Deg,or°. 

In. Ft. 

12 = 1 Yd. 

36 = 3 = 1 PI. 

198 = 16i = 5J = 1 Fur. 
7920 = 660 = 220 = 40 = 1 Mile. 
63360 = 5280 = 1760 = 320 = 8 = 1 

CLOTH MEASURE. 

2 Inches and a quarter.... make 1 Nail marked Nl. 

4 Nails — 1 Quarter of a Yard.. — Qr. 

3 Quarters — 1 Ell Flemish — E F. 

4 Quarters — 1 Yard — Yd. 

5 Quarters — 1 Ell English — EE. 

4Qrs.liInch — 1 Ell Scotch — E S. 

SQUARE MEASURE. 

144 Square Inches make 1 Sq. Foot marked Ft. 

9 Square Feet — 1 Sq. Yard — Yd. 

301 Square Yards — 1 Sq. Pole — Pole. 

40 Square Poles — IRood — Bd. 

4 Boods — 1 Acre — Acr. 

Sq. Inc. Sq. Ft. 

144 = 1 Sq. Yd. 

1296 = 9 = 1 Sq. PI. 

39204 = 2721 = 30i = 1 Bd. 
1568160 = 10890 = 1210 = 40 = 1 Acr. 
6272640 = 43560 = 4840 = 160 = 4 = 1 

When three dimensions are concerned, namely, length, breadth, 
and depth or thickness, it is called cubic or solid measure, which is 
used to measure Timber, Stone, &c. 

The cubic or solid Foot, which is 12 inches in length, and breadth, 
and thickness, contains 1728 cubic or solid inches, and 27 solid 
feet make one solid yard. 



THE PRACTICAL MODEL CALCULATOR. 



DRY, OR CORN MEASURE. 



2 Pints make 1 Quart 



2 Quarts .... 
2 Pottles.... 
2 Gallons.... 

4 Pecks 

8 Bushels... 

5 Quarters... 
2 Weys 

Pts. 

8 = 

16 = 

64 = 

512 = 

2560 = 

5120 = 



Gal. 
1 

2 = 

8 = 

64 = 

320 = 

640 = 



Pec. 
1 

4 = 

32 = 

160 = 

320 = 



marked Qt. 
— Pot. 



1 Pottle — 

1 Gallon — Gal. 

IPeck — Pec. 

1 Bushel — Bu. 

1 Quarter — Qr. 

1 Weigh or Load... — Wey. 

1 Last — Last. 



Bu. 
1 

8 = 
40 = 
80 = 



Qr. 

1 Wey. 

5 = 1 Last. 
10 = 2 = 1 



WINE MEASURE. 



2 Pints make 1 Quart 



2 Quarts 

42 Gallons 

63 Gallons or 

2 Tierces 

2 Hogsheads. 

2 Pipes 



IJ 



Tier.. 



1 Gallon 

1 Tierce 

1 Hogshead 

1 Puncheon 

1 Pipe or Butt. 
ITun 



.marked Qt. 

. — Gal. 

. — Tier. 

. — Hhd. 

. — Pun. 

, — Pi. 

. — Tun. 



Qts. 

1 Gal. 

4 = 1 Tier. 
168 = 42 = 1 Hhd. 
252 = 63 = IJ = 1 Pun. 
336 = 84 = 2 = 11 = 1 Pi. 
504 = 126 = 3 =2 = 1* = 1 Tun. 



Pts. 

2 = 

8 = 

336 = 

504 = 

672 = 

1008 = 

2016 = 1008 = 252 = 6 =4 



3 =2 = 1. 



ALE AND BEER MEASURE. 

2 Pints make 1 Quart marked Qt. 

- 1 Gallon — Gal. 

- 1 Barrel — Bar. 

- 1 Hogshead — Hhd. 

- 1 Puncheon — Pun. 

- IButt — Butt. 

- ITun — Tun. 



4 Quarts 
36 Gallons 

1 Barrel and a half. . 

2 Barrels 

2 Hogsheads 

2 Butts 



Pts. Qt. 

2 = 1 Gal. 

8 = 4 = 1 Bar. 
288 = 144 = 36 = 1 Hhd. 
432 = 216 = 54 = IJ = 1 Butt. 
864 = 432 = 108 = 3 =2 = 1 



OF TIME. y 

OF TIME. 

60 Seconds make 1 Minute marked M. or '. 

60 Minutes — 1 Hour — Hr. 

24 Hours — 1 Day — Day. 

7 Days — IWeek — Wk. 

4 Weeks — 1 Month — Mo. 

13Months^lDay,6Hours,|_ -^j^j.^^Year.... - Yr. 
or ODD Days, b Hours, j 

Sec. Min. 

60 = 1 Hr. 

3600 = 60 = 1 Day. 

86400 = 1440 = 24 = 1 Wk. 
604800 = 10080 = 168 = 7=1 Mo. 
2419200 = 40320 = 672 = 28 =4 = 1 
31557600 = 525960 = 8766 = 365i = 1 Year. 
Wk.Da.Hr. Mo. Da. Hr. 
Or 52 1 6 = 13 1 6 = 1 Julian Year. 
Da. Hr. M. Sec. 
But 365 5 48 48 = 1 Solar Year. 
The time of rotation of the earth on its axis is called a sidereal 
day, for the following reason : If a permanent object be placed on 
the surface of the earth, always retaining the same position, it may 
be so located as to be posited in the same plane with the observer 
and some selected fixed star at the same instant of time ; although 
this coincidence may be but momentary, still this coincidence con- 
tinually recurs, and the interval elapsed between two consecutive 
coincidences has always throughout all ages appeared the same. 
It is this interval that is called a sidereal day. 
The sidereal day increased in a certain ratio, and called the 
mean solar day, has been adopted as the standard of time. 

Thus, 366-256365160 sidereal days = 366-256365160 - 1 or 
365-256365160 mean solar days, whence sidereal day : mean solar 
day : : 365-256365160 : 366-256365160 : : 0-997269672 : 1 or as 
1 : 1-002737803, when 23 hours, ^ minutes 4-0996608 sec. of 
mean solar time = 1 sidereal day; and 24 hours, 3 minutes, 
56-5461797 see. of sidereal time = 1 mean solar day. 

The true solar day is the interval between two successive coinci- 
dences of the sun with a fixed object on the earth's surface, bring- 
ing the sun, the fixed object, and the observer in the same plane. 

This interval is variable, but is susceptible of a maximum and 
minimum, and oscillates about that mean period which is called a 
mean solar day. 

Apparent or true time is that which is denoted by the sun-dial, 
from the apparent motion of the sun in its diurnal revolution, and 
difi'ers several minutes in certain parts of the ecliptic from the 
mean time, or that shown by the clock. The difi'erence is called 
the equation of time, and is set down in the almanac, in order to 
ascertain the true time. 



ARITHMETIC. 



Arithmetic is the art or science of numbering; being that 
branch of Mathematics which treats of the nature and properties 
of numbers. When it treats of whole numbers, it is called Com- 
mon Arithmetic ; but when of broken numbers, or parts of num- 
bers, it is called Fractions. 

Unity, or a Unit, is that by which every thing is called one ; 
being the beginning of number ; as one man, one ball, one gun. 

Number is either simply one, or a compound of several units ; 
as one man, three men, ten men. 

An Integer or Whole Number, is some certain precise quantity 
of units ; as one, three, ten. These are so called as distinguished 
from Fractions, which are broken numbers, or parts of numbers ; 
as one-half, two-thirds, or three-fourths. 

NOTATION AND NUMERATION. 

Notation, or Numeration, teaches to denote or express any pro- 
posed number, either by words or characters ; or to read and write 
down any sum or number. 

The numbers in Arithmetic are expressed by the following ten 
digits, or Arabic numeral figures, which were introduced into 
Europe by the Moors about eight or nine hundred years since : 
viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 
9 nine, cipher or nothing. These characters or figures were 
formerly all called by the general name of Ciphers; whence it 
came to pass that the art of Arithmetic was then often called 
Ciphering. Also, the first nine are called Significant Figures, as 
distinguished from the cipher, which is quite insignificant of itself. 

Besides this value of those figures, they have also another, which 
depends upon the place they stand in when joined together ; as in 
the following Table : 



&c. 



o 

1 


o 


o 

;=! 


o 

2 

■73 

a 

W 


.13 
H 

o 


i 

O 




J 


•a 


9 


8 


7 


6 


5 


4 


3 


2 


1 




9 


8 


7 


6 


5 


4 


3 


2 






9 


8 


7 


6 


5 


4 


3 








9 


8 


7 


6 


5 


4 










9 


8 
9 


7 
8 
9 


6 

7 
8 
9 


5 
6 
7 
8 
9 



10 



NOTATION AND NUMERATION. 11 

Here any figure in the first place, reckoning from right to left, 
denotes only its own simple value ; but that in the second place 
denotes ten times its simple value ; and that in the third place 
a hundred times its simple value ; and so on ; the value of any 
figure, in each successive place, being always ten times its former- 
value. 

Thus, in the number 1796, the 6 in the first place denotes only 
six units, or simply six ; 9 in the second place signifies nine tens, 
or ninety : 7 in the third place, seven hundred ; and the 1 in the 
fourth place, one thousand ; so that the whole number is read thus — 
one thousand seven hundred and ninety-six. 

As to the cipher 0, it stands for nothing of itself, but being 
joined on the right-hand side to other figures, it increases their 
value in the same tenfold proportion : thus, 5 signifies only five ; 
but 50 denotes 5 tens, or fifty; and 500 is five hundred; and 
so on. 

For the more easily reading of large numbers, they are divided 
into periods and half-periods, each half-period consisting of three 
figures ; the name of the first period being units ; of the second, 
millions ; of the third, millions of millions, or bi-millions, contracted 
to billions ; of the fourth, millions of millions of millions, or tri- 
millions, contracted to trillions ; and so on. Also, the first part 
of any period is so many units of it, and the latter part so many 
thousands. 

The following Table contains a summary of the whole doc- 
trine : 



Periods. 


QuadrilL; 


Trillions; Billions; Millions; Units. 


Half-per. 


th. un. 


th. un. th. un. th. un. th. un. 


Figures. 


123,456; 


789,098; 765,432; 101,234; 567,890. 



Numeration is the reading of any number in words that is pro- 
posed or set down in figures. 

Notation is the setting down in figures any number proposed in 
words. 

OF THE R0MA:N NOTATION. 

The Eomans, like several other nations, expressed their numbers 
by certain letters of the alphabet. The Romans only used seven 
numeral letters, being the seven following capitals : viz. I for one ; 
V for jive ; X for ten ; L for fifty ; C for a hundred ; D for five hun- 
dred ; M for a thousand. The other numbers they expressed by 
various repetitions and combinations of these, after the following 
manner : 



12 THE PRACTICAL MODEL CALCULATOR. 

1=1. 

2 = II. As often as any character is repeated, 

3 = III. so many times is its value repeated. 

4 = IIII. or IV. A less character before a greater 

5 = V. diminishes its value. 

6 = VI. A less character after a greater in- 

7 = VII. creases its value. 

8 = VIII. 

9 = IX. 
10 = X. 
50 = L. 

100 = C. 

500 = D or ID, ^qj. every annexed, this becomes 

ten times as many. 
1000 = M or CIO. For every C and 0, placed one at each 

2000 = MM. end, it becomes ten times as much. 

5000 = V o r 100. A bar over any number increases it 

6000 = VL 1000 fold. 

10000 = Tor CCIOO. 
50000 = L or IQOO. 
60000 = L_X. 
100000 = C_or CCCIOOO. 
1000000 = Mor CCCCIOOOO. 
2000000 = MM. 
&c. &c. 

EXPLANATION OP CERTAIN CHARACTERS. 

There are various characters or marks used in Arithmetic and 
Algebra, to denote several of the operations and propositions ; the 
chief of which are as follow : 



: :: : proportion. 

= equality. 

>/ square root. 

■^ cube root, &c. 



-f signifies plus, or addition. 

— minus, or subtraction. 

X multiplication. 

-T- division. 

Thus, 

5 + 3, denotes that 3 is to be added to 5 = 8. 

6 — 2, denotes that 2 is to be taken from 6 = 4. 
7x3, denotes that 7 is to be multiplied by 3 = 21. 
8 -T- 4, denotes that 8 is to be divided by 4 = 2. 

2 : 3 :: 4 : 6, shows that 2 is to 3 as 4 is to 6, and thus, 2x6=3x4. 
6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. 

s/S, or 3^, denotes the square root of the number 3 = 1*7320508. 

-^5, or 5*, denotes the cube root of the number 5 = 1*709976. 
7^ denotes that the number 7 is to be squared = 49. 
8^, denotes that the number 8 is to be cubed = 512. 
&c. 



RULE OF THREE. 13 

RULE OF THREE. 

The Rule of Three teaches how to find a fourth proportional 
to three numbers given. Whence it is also sometimes called the 
Kule of Proportion. It is called the Rule of Three, because three 
terms or numbers are given to find the fourth ; and because of its 
great and extensive usefulness, it is often called the Golden Rule. 

This Rule is usually considered as of two kinds, namely, Direct 
and Inverse. 

The Rule of Three Direct is that in which more requires more, or 
less requires less. As in this : if 3 men dig 21 yards of trench in 
a certain time, how much will 6 men dig in the same time ? Here 
more requires more, that is, 6 men, which are more than 3 men, 
will also perform more work in the same time. Or when it is thus : 
if 6 men dig 42 yards, how much will 3 men dig in the same time ? 
Here, then, less requires less, or 3 men will perform proportionally 
less work than 6 men in the same time. In both these cases, then, 
the Rule, or the Proportion, is Direct; and the stating must be 
thus. As 3 : 21 : : 6 : 42, 
or thus. As 6 : 42 : : 3 : 21. 

But, the Rule of Three Inverse is when more requires less, or 
less requires more. As in this : if 3 men dig a certain quantity 
of trench in 14 hours, in how many hours will 6 men dig the like 
quantity? Here it is evident that 6 men, being more than 3, will 
perform an equal quantity of work in less time, or fewer hours. 
Or thus : if 6 men perform a certain quantity of work in 7 hours, 
in how many hours will 3 men perform the same ? Here less 
requires more, for 3 men will take more hours than 6 to perform 
the same work. In both these cases, then, the Rule, or the Pro- 
portion, is Inverse ; and the stating must be 
thus. As 6 : 14 : ; 3 : 7, 
or thus, As 3 : 7 : : 6 : 14. 

And in all these statings the fourth term is found, by multiply- 
ing the 2d and 3d terms together, and dividing the product by the 
1st term. 

Of the three given numbers, two of them contain the supposi- 
tion, and the third a demand. And for stating and working ques- 
tions of these kinds observe the following general Rule : 

Rule. — State the question by setting down in a straight line the 
three given numbers, in the following manner, viz. so that the 2d 
term be that number of supposition which is of the same kind that 
the answer or 4th term is to be ; making the other number of sup- 
position the 1st term, and the demanding number the 3d term, 
when the question is in direct proportion ; but contrariwise, the 
other number of supposition the third term, and the demanding 
number the 1st term, when the question has inverse proportion. 

Then, in both cases, multiply the 2d and 3d terms together, and 
divide the product by the first, which will give the answer, or 4th 
term sought, of the same denomination as the second term. 
B 



14 THE PRACTICAL MODEL CALCULATOR. 

Note, If the first and third terms consist of different denomina- 
tions, reduce them both to the same ; and if the second term be a 
compound number, it is mostly convenient to reduce it to the lowest 
denomination mentioned. If, after division, there be any remainder, 
reduce it to the next lower denomination, and divide by the same 
divisor as before, and the quotient will be of this last denomina- 
tion. Proceed in the same manner with all the remainders, till 
they be reduced to the lowest denomination which the second term 
admits of, and the several quotients taken together will be the 
answer required. 

Note also, The reason for the foregoing Kules will appear when 
we come to treat of the nature of Proportions. Sometimes also 
two or more statings are necessary, which may always be known 
from the nature of the question. 

An engineer having raised 100 yards of a certain work in 
24 days with 5 men, how many men must he employ to finish a 
like quantity of work in 15 days ? 

da. men. da. men. 
As 15 : 5 : : 24 : 8 Ans. 
5 
15 ) 120 ( 8 Answer. 
120 



COMPOUND PROPORTION. 

Compound Proportion teaches how to resolve such questions as 
require two or more statings by Simple Proportion; and that, 
whether they be Direct or Inverse. 

In these questions, there is always given an odd number of terms, 
either five, or seven, or nine, &c. These are distinguished into 
terms of supposition and terms of demand, there being always one 
term more of the former than of the latter, which is of the same 
kind with the answer sought. 

Rule. — Set down in the middle place that term of supposition 
which is of the same kind with the answer sought. Take one of 
the other terms of supposition, and one of the demanding terms 
which is of the same kind with it ; then place one of them for a 
first term, and the other for a third, according to the directions 
given in the Rule of Three. Do the same with another term of 
supposition, and its corresponding demanding term ; and so on if 
there be more terms of each kind ; setting the numbers under each 
other which fall all on the left-hand side of the middle term, and 
the same for the others on the right-hand side. Then to work. 

By several Operations. — Take the two upper terms and the mid- 
dle term, in the same order as they stand, for the first Rule of 
Three question to be worked, whence will be found a fourth term. 
Then take this fourth number, so found, for the middle term of a 
second Rule of Three question, and the next two under terms in the 
general stating, in the same order as they stand, finding a fourth 



OF COMMON FRACTIONS. 15 

term from them ; and so on, as far as there are any numbers in the 
general stating, making always the fourth number resulting from 
each simple stating to be the second term of the next following one. 
So shall the last resulting number be the answer to the question. 

By one Operation. — Multiply together all the terms standing 
under each other, on the left-hand side of the middle term ; and, in 
like manner, multiply together all those on the right-hand side of 
it. Then multiply the middle term by the latter product, and 
divide the result by the former product, so shall the quotient be 
the answer sought. 

How many men can complete a trench of 135 yards long in 
8 days, when 16 men can dig 54 yards in 6 days ? 

G-eneral 
yds. 54 : 16 men : : 135 yds. 
days 8 6 days 

432 810 

_16 

4860 
810 



432) 12960 (30 Ans. by one operation. 
1296 




1st. 
As 54 : 16 : : 135 ; 40 



The same hy two operations. 

2d. 

As 8 : 40 : : 6 : 30 

6 



_16 

810 
135 
54)2160(40 
216 



8 ) 240 ( 30 Ans. 
24 



OF COMMON FRACTIONS. 

A Fraction, or broken number, is an expression of a part, or 
some parts, of something considered as a whole. 

It is denoted by two numbers, placed one below the other, with 
a line between them : 

, 3 numerator ) 
^^''''Tdenominator / ^^^^^ ^^ ^^^^^ three-fourths. 

The Denominator, or number placed below the line, shows how 
many equal parts the whole quantity is divided into ; and repre- 
sents the Divisor in Division. And the Numerator, or number set 
above the line, shows how many of those parts are expressed by the 
Fraction; being the remainder after division. Also, both these 
numbers are, in general, named the Terms of the Fractions. 



16 THE PRACTICAL MODEL CALCULATOR. 

Fractions are either Proper, Improper, Simple, Compound, or' 
Mixed. 

A Proper Fraction is when the numerator is less than the deno- 
minator; as J, or f, or f, &c. 

An Improper Fraction is when the numerator is equal to, or 
exceeds, the denominator; as |, or {, or f, &c. 

A Simple Fraction is a single expression denoting any number 
of parts of the integer ; as f , or |. 

A Compound Fraction is the fraction of a fraction, or several 
fractions connected with the word of between them ; as J of f , or 
I of f of 3, &c. 

A Mixed Number is composed of a whole number and a fraction 
together; as 3 J, or 12 1, &c. 

A whole or integer number may be expressed like a fraction, by 
writing 1 below it, as a denominator ; so 3 is f, or 4 is |, &c. 

A fraction denotes division ; and its value is equal to the quo- 
tient obtained by dividing the numerator by the denominator ; 
so '^f is equal to 3, and ^/ is equal to 4. 

Hence, then, if the numerator be less than the denominator, the 
value of the fraction is less than 1. If the numerator be the same 
as the denominator, the fraction is just equal to 1. And if the 
numerator be greater than the denominator, the fraction is greater 
than 1. 

REDUCTION OF FRACTIONS. 

Reduction of Fractions is the bringing them out of one form or 
denomination into another, commonly to prepare them for the opera- 
tions of Addition, Subtraction, &c., of which there are several cases. 

To find the greatest common measure of tivo or more numbers. 

The Common Measure of two or more numbers is that number 
which will divide them both without a remainder : so 3 is a com- 
mon measure of 18 and 24 ; the quotient of the former being 6, 
and of the latter 8. And the greatest number that will do this, 
is the greatest common measure : so 6 is the greatest common mea- 
sure of 18 and 24 ; the quotient of the former being 3, and of the 
latter 4, which will not both divide farther. 

KuLE. — If there be two numbers only, divide the greater by 
the less ; then divide the divisor by the remainder ; and so on, divid- 
ing always the last divisor by the last remainder, till nothing 
remains ; then shall the last divisor of all be the greatest common 
measure sought. 

When there are more than two numbers, find the greatest com- 
mon measure of two of them, as before ; then do the same for that 
common measure and another of the numbers ; and so on, through 
all the numbers ; then will the greatest common measure last found 
be the answer. 

If it happen that the common measure thus found is 1, then the 
numbers are said to be incommensurable, or to have no common 
measure. 



REDUCTION OF FRACTIONS. 17 

To find the greatest common measure of 1998, 918, and 522. 
918 ) 1998 (2 So 54 is the greatest common measure 

1836 of 1998 and 918. 

~162 ) 918 ( 5 Hence 54 ) 522 ( 9 
810 486 

108)162(1 36)54(1 

108 36 

54)108(2 18)36(2 

108 36 

So that 18 is the answer required. 

To abbreviate or reduce fractions to their loivest terms. 

Rule. — Divide the terms of the given fraction by any number 
that will divide them without a remainder ; then divide these quo- 
tients again in the same manner ; and so on, till it appears that 
there is no number greater than 1 which will divide them ; then the 
fraction will be in its lowest terms. 

Or, divide both the terms of the fraction by their greatest com- 
mon measure, and the quotients will be the terms of the fraction, 
required, of the same value as at first. 

That dividing both the terms of the fraction by the same num- 
ber, whatever it be, will give another fraction equal to the former,, 
is evident. And when those divisions are performed as often as 
can be done, or when the common divisor is the greatest possible, 
the terms of the resulting fraction must be the least possible. 

1. Any number ending with an even number, or a cipher, is divi- 
sible, or can be divided by 2. 

2. Any number ending with 5, or 0, is divisible by 5. 

3. If the right-hand place of any number be 0, the whole is 
divisible by 10 ; if there be 2 ciphers, it is divisible by 100 ; if 
3 ciphers, by 1000; and so on, which is only cutting off those 
ciphers. 

4. If the two right-hand figures of any number be divisible 
by 4, the whole is divisible by 4. And if the three right-hand 
figures be divisible by 8, the whole is divisible by 8 ; and so on. 

5. If the sum of the digits in any number be divisible by 3, or 
by 9, the whole is divisible by 3, or by 9. 

6. If the right-hand digit be even, and the sum of all the digits 
be divisible by 6, then the whole will be divisible by 6. 

7. A number is divisible by 11 when the sum of the 1st, 3d, 
5th, &c., or of all the odd places, is equal to the sum of the 2d, 
4th, 6th, &c., or of all the even places of digits. 

8. If a number cannot be divided by some quantity less than 
the square of the same, that number is a prime, or cannot be 
divided by any number whatever. 

9. All prime numbers, except 2 and 5, have either 1, 3, 7, or 9, 
in the place of units j and all other numbers are composite, or can. 
be divided. 

b2 2 



18 THE PRACTICAL MODEL CALCULATOR. 

10. When numbers, with a sign of addition or subtraction between 
them, are to be divided by any number, then each of those num- 

10 + 8-4 
bers must be divided by it. Thus, ^ =5 +4 — 2 = 7. 

11. But if the numbers have the sign of multiplication between 

. -,..•. 1 m, 10 x8 X 3 
them, only one of them must be divided. Thus, — ^ ^ = 

10x4x3 10x4x1 10x2x1 20 
6x1 ~ 2x1 " 1x1 -jL"' 
JReduce ||f to its least terms. 

Mf = ^ = f!= if = I'a = I? the answer. 
Or tlius : 

144 ) 240 ( 1 Therefore 48 is the greatest common measure, and 
144 48 ) i|^ = f the answer, the same as before. 

"96)144(1 
_96 

48)96(2 
_96 

To reduce a mixed number to its equivalent improper fraction. 
EuLE. — ^Multiply the whole - number by the denominator of the 
fraction, and add the numerator to the product ; then set that sum 
above the denominator for the fraction required. 
Beduce 23f to a fraction. 

Or, 
23 (23 X 5) + 2 117 
5 5 ~ 5 • 

115 
2 

117 
5 

To reduce an improper fraction to its equivalent whole or mixed 

number. 
Rule. — Divide the numerator by the denominator, and the quo- 
tient will be the whole or mixed number sought. 

Reduce ^f to its equivalent number. 

Here V or 12 ^ 3 = 4. 
Reduce \^ to its equivalent number. 

Here \' or 15 -r- 7 = 2f 
Reduce ''■^ to its equivalent number. 
Thus, 17) 749 (44J^ 
68_ 

69 Sothat '^9 =44j\ 



REDUCTION OF FRACTIONS. 19 

To reduce a ivTiole numher to an equivalent fraction, having a 

given denominator. 
Rule. — Multiply the whole number by the given denominator, 
then set the product over the said denominator, and it will form 
the fraction required. 

Reduce 9 to a fraction whose denominator shall be 7. 

Here 9 x 7 = 63, then ^^ is the answer. 
For 6^3 = 63 -H 7 = 9, the proof. 

To reduce a compound fraction to an equivalent simple one. 

Rule. — Multiply all the numerators together for a numerator, 
and all the denominators together for the denominator, and they 
will form the simple fraction sought. 

When part of the compound fraction is a whole or mixed number, 
it must first be reduced to a fraction by one of the former cases. 

And, when it can be done, any two terms of the fraction may be 
divided by the same number, and the quotients used instead of 
them. Or, when there are terms that are common, they may be 
omitted. 

Reduce J of f of | to a simple fraction. 

1x2x3 6 _ 1 

^®^® 2x3x4" 24 ~ 4* 
^1x2x3 1, .. , 

Or, Q Q . = 2? hy omitting the twos and threes. 

Reduce | of f of ^ to a simple fraction. 

2 X 3 X 10 60 12 4 
^^^^ 3 X 5 X 11 ~ 165 ~ 33 ~ 11* 
^ 2 X 3 X 10 4 , 
Or, 3 X 5 X 11 ~ 11' ^^^® ^^ before. 

To reduce fractions of different denominators to equivalent frac- 
tions, having a common denominator. 

Rule. — Multiply each numerator into all the denominators ex- 
cept its own for the new numerators ; and multiply all the denomi- 
nators together for a common denominator. 

It is evident, that in this and several other operations, when any 
of the proposed quantities are integers, or mixed numbers, or com- 
pound fractions, they must be reduced, by their proper rules, to 
the form of simple fractions. 

Reduce J, f , and f to a common denominator. 

1 X 3 X 4 = 12 the new numerator for J. 

2x2x4 = 16 for f. 

3x2x3 = 18 for |. 

2 X 3 X 4 = 24 the common denominator. 
Therefore, the equivalent-fractions are ^, |f, and If. 

Or, the whole operation of multiplying may be very well per- 
formed mentally, and only set down the results and given fractions 
thus :},!,! = M) M? M = 12? 12? ^? V abbreviation. 



20 THE PRACTICAL MODEL CALCULATOR. 

When the denominators of two given fractions have a common 
measure, let them be divided by it ; then multiply the terms of 
each given fraction by the quotient arising from the other's deno- 
minator. 

When the less denominator of two fractions exactly divides the 
greater, multiply the terms of that which hath the less denominator 
by the quotient. 

When more than two fractions are proposed, it is sometimes con- 
venient first to reduce two of them to a common denominator, 
then these and a third ; and so on, till they be all reduced to their 
least common denominator. 

TofindtTie value of a fraction in parts of the integer. 

Rule. — Multiply the integer by the numerator, and divide the 
product by the denominator, by Compound Multiplication and 
Division, if the integer be a compound quantity. 

Or, if it be a single integer, multiply the numerator by the parts 
in the next inferior denomination, and divide the product by the 
denominator. Then, if any thing remains, multiply it by the parts 
in the next inferior denomination, and divide by the denominator 
as before ; and so on, as far as necessary ; so shall the quotients, 
placed in order, be the value of the fraction required. 

What is the value of | of a pound troy ? 7 oz. 4 dwts. 

What is the value of ^ of a cwt.? 1 qr. 7 lb. 

What is the value of f of an acre ? 2 ro. 20 p.o. 

What is the value of ^^ of a day ? 7 hrs. 12 min. 

To reduce a fraction from one denomination to another. 

Rule. — Consider how many of the less denomination make 
one of the greater; then multiply the numerator by that num- 
ber, if the reduction be to a less name, or the denominator, if to 
a greater. 

Reduce f of a cwt. to the fraction of a pound. 

I X f X 2-8 = U, 

ADDITION OF FRACTIONS. 
To add fractions togetlier that have a common denominator. 

Rule. — Add all the numerators together, and place the sum 
over the common denominator, and that will be the sum of the 
fractions required. 

If the fractions proposed have not a common denominator, they 
must be reduced to one. Also, compound fractions must be reduced 
to simple ones, and mixed numbers to improper fractions; also, 
fractions of different denominations to those of the same denomi- 
nation. 

To add f and | together. - Here | -f f = | = If 

To add I and f together.^ | + 5=m + |i = m = ii|. 

To add I and 7^ and f of f together. 

5 4- fl -U 1 of 3 = 5 4- 1_5 -I- 1 = 5 _L 60 4- 2 = 6_7 = g3 
8''2~3^-'-4 8'2'4 8'8~8 8 *^&* 



RULE OF THREE IN FRACTIONS. M. 

SUBTRACTION OF FRACTIONS. 

Rule. — Prepare the fractions the same as for Addition ; then sub- 
tract the one numerator from the other, and set the remainder over 
the common denominator, for the difference of the fractions sought. 

To find the difference between f and f 

TTpvp 5 _ 1 = 4 = 2 
iXCiC 6 6 6 3* 

To find the difference between f and I. 

3 5 = 21 20=i 

1 T 28 28 28* 

MULTIPLICATION OF FRACTIONS. 

Multiplication of any thing by a fraction implies the taking 
some part or parts of the thing ; it may therefore be truly expressed 
by a compound fraction ; which is resolved by multiplying together 
the numerators and the denominators. 

Rule, — Reduce mixed numbers, if there be any, to equivalent 
fractions ; then multiply all the numerators together for a nume- 
rator, and all the denominators together for a denominator, which 
will give the product required. 

Required the product of | and |. 

TTavp 3 y 2 — - 6 — 1 

Or, f X I = I X i = 1. 

Required the continued product of f, 3|, 5, and f of f . 

2 13 5 3 3 13 X 3 39 
Here gX-^XjX^Xg^ ^^^ = "s^ = 4|. 

DIVISION OF FRACTIONS. 
Rule. — Prepare the fractions as before in Multiplication ; then 
divide the numerator by the numerator, and the denominator by 
the denominator, if they will exactly divide ; but if not, then invert 
the terms of the divisor, and multiply the dividend by it, as in 
Multiplication. 

Divide ^^ by |. 

Here f -^ | = | = If, by the first method. 
Divide | by ^. 

Here f -- .-^^ = | x i^^ = | X | = f = 4i, by the latter. 

RULE OF THREE IN FRACTIONS. 
Rule. — Make the necessary preparations as before directed; 
then multiply continually together the second and third terms, and 
the first with its terms inverted as in Division, for the answer. 
This is only multiplying the second and third terms together, and 
dividing the product by the first, as in the Rule of Three in whole 
numbers. 

If f of a yard of velvet cost | of a dollar, what will ^^ of a 

yard cost? 

Tx 3 2 5 8 2 5 , , ,, 

Here ^:-p::r-7^:oXrXY^ = |ofa dollar. 



22 THE PRACTICAL MODEL CALCULATOR. 

DECIMAL FEACTIONS. 

A Decimal Fraction is that which has for its denominator a 
unit (1) with as many ciphers annexed as the numerator has places ; 
and it is usually expressed by setting down the numerator only, 
with a point before it on the left hand. Thus, -^ is '5, and ^ is 
•25, and j^ is -075, and yowoo is '00124 ; where ciphers are pre- 
fixed to make up as many places as are in the numerator, when 
there is a deficiency of figures. 

A mixed number is made up of a whole number with some deci- 
mal fraction, the one being separated from the other by a point. 
Thus, 3-25 is the same as 3^^, or fff. 

Ciphers on the right hand of decimals make no alteration in 
their value; for '5, or '50, or '500, are decimals having all the 
same value, being each = i^ or ^. But if they are placed on the 
left hand, they decrease the value in a tenfold proportion. Thus, 
•5 is ^ or 5 tenths, but -05 is only ^fo o^ ^ hundreths, and -005 
is but YMo or 5 thousandths. 

The first place of decimals, counted from the left hand towards 
the right, is called the place of primes, or lOths ; the second is the 
place of seconds, or lOOths ; the third is the place of thirds, or 
lOOOths; and so on. For, in decimals, as well as in whole num- 
bers, the values of the places increase towards the left hand, and 
decrease towards the right, both in the same tenfold proportion ; 
as in the following Scale or Table of Notation: 



5 " 

rS 



13 u 

_- ^, O cS 






3333333 333333 

addition of decimals. 
Rule. — Set the numbers under each other according to the value 
of their places, like as in whole numbers ; in which state the deci- 
mal separating points will stand all exactly under each other. 
Then, beginning at the right hand, add up all the columns of 
number as in integers, and point off as many places for decimals as 
are in the greatest number of decimal places in any of the lines that 
are added ; or, place the point directly below all the other points. 
To add together 29-0146, and 3146-5, 29-0146 

and 2109, and 62417, and 14-16. 3146-5 

2109- 

•62417 
14-16 

5299-29877, the sum. 



MULTIPLICATION OF DECIMALS. 23 

The sum of 376-25 + 86-125 + 637*4725 + 6-5 + 41-02 + 
358-865 = 1506.2325. 
• The sum of 3-5 + 47-25 + 2.0073 + 927-01 + 1-5 = 981.2673. 

The sum of 276 + 54-321 + 112 + 0.65 + 12-5 + '0463 = 
455-5173. 

SUBTRACTION OF DECIMALS. 

EuLE. — Place the numbers under each other according to the 
value of their places, as in the last rule. Then, beginning at the 
right hand, subtract as in whole numbers, and point off the deci- 
mals as in Addition. 

To find the difference between 91-73 
91.73 and 2.138. 2-138 

89-592 the difference. 

The difference between 1-9185 and 2-73 = 0-8115. 

The difference between 214-81 and 4-90142 = 209-90858. 

The difference between 2714 and -916 = 2713-084. 

MULTIPLICATION OF DECIMALS. 
Rule. — Place the factors, and 
multiply them together the same 
as if they were whole numbers. 
Then point off in the product just 
as many places of decimals as 
there are decimals in both the fac- 
tors. But if there be not so many 
figures in the product, then supply 
the defect by prefixing ciphers. 

Multiply 79-347 by 23-15, and we have 1836-88305. 
Multiply -63478 by -8204, and we have -520773512. 
Multiply -385746 by -00464, and we have -00178986144. 

CONTRACTION I. 

To multiple/ decimals hy 1 ivith any number of ciphers, as 10, or 
100, or 1000, ^^c. 
This is done by only removing the decimal point so many places 
farther to the right hand as there are ciphers in the multiplier ; 
and subjoining ciphers if need be. 

The product of 51-3 and 1000 is 51300. 
The product of 2-714 and 100 is 271-4. 
The product of -916 and 1000 is 916. 
The product of 21-31 and 10000 is 213100. 

CONTRACTION II. 

To contract the operation, so as to retain only as many decimals in 
the product as may he thought necessary, ivhen the product ivould 
naturally contain several more places. 
Set the units' place of the multiplier under that figure of the 

multiplicand whose place is the same as is to be retained for the 



Multiply -321096 
by -2465 
1605480 
1926576 
1284384 
642192 
•0791501640 the product. 



24 THE PRACTICAL MODEL CALCULATOR. 

last in tlie product ; and dispose of the rest of tlie figures in the 
inverted or contrary order to what they are usually placed in. 
Then, in multiplying, reject all the figures that are more to the 
right than each multiplying figure ; and set down the products, so 
that their right hand figures may fall in a column straight below 
each other ; but observing to increase the first figure of every line 
with what would arise from the figures omitted, in this manner, 
namely, 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c. ; 
and the sum of all the lines will be the product as required, com- 
monly to the nearest unit in the last figure. 

To multiply 27-14986 by 92-41035, so as to retain only four 
places of decimals in the product. 

Contracted wav. Common way. 

27-14986 " 27-14986 

53014-29 92-41035 



24434874 

542997" 

108599 

2715 

81 

14 

2508-9280 



13 


574930 


81 


'44958 


2714 


986 


108599 


44 


542997, 


2 


24434874 


1 


2508-9280 


650510 



DIVISION OF DECIMALS. 

KuLE. — Divide as in whole numbers ; and point off in the quo- 
tient as many places for decimals, as the decimal places in the 
dividend exceed those in the divisor. 

When the places of the quotient are not so many as the rule re- 
quires, let the defect be supplied by prefixing ciphers. 

Y^hen there happens to be a remainder after the di-^dsion ; or 
when the decimal places in the divisor are more than those in the 
dividend ; then ciphers may be annexed to the dividend, and the 
quotient carried on as far as requii'ed. 

179 ) -48624097 ( -00271643 | -2685 ) 27-00000 ( 100-55865 

1282 15000 

294 15750 

1150 23250 

769 17700 

537 15900 

000 24750 

Divide 234-70525 by 64-25. 3-653. 

Divide 14 by -7854. 17-825. 

Divide 2175-68 by 100. 21-7568. 

Divide -8727587 by -162. 5-38739. 

CONTRACTION I. 

"When the divisor is an integer, with any number of ciphers an- 
nexed ; cut off those ciphers, and remove the decimal point in the 



REDUCTION OF DECIMALS. 25 

dividend as many places farther to the left as there are ciphers cut 
off, prefixing ciphers if need be ; then proceed as before. 
Divide 45-5 hj 2100. 

21-00) 455 (-0216, &c. 
35 
140 
14 

CONTRACTION n. 

Hence, if the divisor be 1 with ciphers, as 10, or 100, or 1000, 
&c. ; then the quotient will be found by merely moving the decimal 
point in the dividend so many places farther to the left as the di- 
visor has ciphers ; prefixing ciphers if need be. 

So, 217-3 -4- 100 = 2-1T3, and 419 ^ 10 = 41-9. 
And 5-16 -i- 100 = -0516, and -21 -i- 1000 = -00021. 

CONTRACTION HI. 

When there are many figures in the divisor ; or only a certain 
number of decimals are necessary to be retained in the quotient, 
then take only as many figures of the divisor as will be equal to 
the number of figures, both integers and decimals, to be in the quo- 
tient, and find how many times they may be contained in the first 
figures of the dividend, as usual. 

Let each remainder be a new dividend ; and for every such divi- 
dend, leave out one figure more on the right hand side of the di- 
visor ; remembering to carry for the increase of the figures cut off, 
as in the 2d contraction in Multiplication. 

When there are not so many figures in the divisor as are required 
to be in the quotient, begin the operation with all the figures, and 
continue it as usual till the number of figures in the divisor be equal 
to those remaining to be found in the quotient, after which begin 
the contraction. 

Divide 2508-92806 by 92-41035, so as to have only four deci- 
mals in the quotient, in which case the quotient will contain six 
figures. 

Contracted. Common way. 



92-4103,5) 2508-928,06 (27-1498 

660721 

13849 

4608 

912 

80 

6 



92-4103,5) 2508-928,06 (27*1498 
66072106 
13848610 
46075750 
91116100 
79467850 
5539570 



REDUCTION OF DECIMALS. 
To reduce a common fraction to its equivalent decimal. 
Rule. — Divide the numerator by the denominator as in Division 
of Decimals, annexing ciphers to the numerator as far as necessary; 
so shall the quotient be the decimal required. 



26 THE PRACTICAL MODEL CALCULATOR. 

Eeduce ^ to a decimal. 

24 = 4 X 6. Then 4 ) 7- 

6)1-750000 
'291666, &c. 
f reduced to a decimal, is •375. 

■^ reduced to a decimal, is '04. 

3I2 reduced to a decimal, is -015625. 

^^ reduced to a decimal, is -071577, &c. 

CASE n. 
To find the value of a decimal in terms of the inferior denominations. 

Rule. — Multiply the decimal by the number of parts in the next 
lower denomination ; and cut off as many places for a remainder, 
to the right hand, as there are places in the given decimal. 

Multiply that remainder by the parts in the next lower denomi- 
nation again, cutting off for another remainder as before. 

Proceed in the same manner through all the parts of the integer; 
then the several denominations, separated on the left hand, will 
make up the value required. 

What is the value of -0125 lb. troy : — 3 dwts. 

What is the value of -4694 lb. troy:— 5 oz. 12 dwt. 15*744 gr. 

What is the value of -625 cwt. : — 2 qr. 14 lb. 

What is the value of -009943 miles :— 17 yd. 1 ft. 5-98848 in. 

What is the value of -6875 yd. :— 2 qr. 3 nls. 

What is the value of -3375 ac. : — 1 rd. 14 poles. 

What is the value of -2083 hhd. of wine :— 13-1229 gal. 

CASE in. 
To reduce integers or decimals to equivalent decimals of higher 

denominations. 
Rule. — Divide by the number of parts in the next higher de- 
nomination ; continuing the operation to as many higher denomi- 
nations as may be necessary, the same as in Reduction Ascending 
of whole numbers. 

Reduce 1 dwt. to the decimal of a pound troy. 



20 
12 



1 dwt. 
0-05 oz. 
0-004166, &c. lb. 



Reduce 7 dr. to the decimal of a pound avoird.: — -02734375 lb. 

Reduce 2-15 lb. to the decimal of a cwt. : — -019196 cwt. 

Reduce 24 yards to the decimal of a mile: — -013636, &c. miles. 

Reduce -056 poles to the decimal of an acre : — -00035 ac. 

Reduce 1*2 pints of wine to the decimal of a hhd. : — -00238 hhd. 

Reduce 14 minutes to the decimal of a day : — -009722, &c. da. 

Reduce -21 pints to the decimal of a peck: — -013125 pec. 

When there are several numbers, to he reduced all to the decimal of 

the highest. 

Set the given numbers directly under each other, for dividends, 
proceeding orderly from the lowest denomination to the highest. 



DUODECIMALS. 27 

Opposite to each dividend, on the left hand, set such a number 
for a divisor as will bring it to the next higher name ; drawing a 
perpendicular line between all the divisors and dividends. 

Begin at the uppermost, and perform all the divisions ; only ob- 
serving to set the quotient of each division, as decimal parts, on 
the right hand of the dividend next below it ; so shall the last quo- 
tient be the decimal required. 

Reduce 5 oz. 12 dwts. 16 gr. to lbs. :— -46944, &c. lb. 

RULE OF THREE IN DECIMALS. 

Rule. — Prepare the terms by reducing the vulgar fractions to 
decimals, any compound numbers either to decimals of the higher 
denominations, or to integers of the lower, also the first and third 
terms to the same name : then multiply and divide as in whole 
numbers. 

Any of the convenient examples in the Rule of Three or Rule of 
Five in Integers, or Common Fractions, may be taken as proper 
examples to the same rules in Decimals. — The following example, 
which is the first in Common Fractions, is wrought here to show the 
method. 

If f of a yard of velvet cost f of a dollar, what will ^^ yd. cost ? 
yd. $ yd. $ 
I = -375 -375 : -4 : : -3125 : -333, &c. 

•4 



•4 -375 ) -12500 ( -333333, 33i cts. 

1250 
125 



,^ = -3125. 



DUODECIMALS. 



Duodecimals, or Cross Multiplication, is a rule made use of 
by workmen and artificers, in computing the contents of their works. 

Dimensions are usually taken in feet, inches, and quarters ; any 
parts smaller than these being neglected as of no consequence. 
And the same in multiplying them together, or casting up the con- 
tents. 

Rule. — Set down the two dimensions, to be multiplied together, 
one under the other, so that feet stand under feet, inches under 
inches, &c. 

Multiply each term in the multiplicand, beginning at the lowest, 
by the feet in the multiplier, and set the result of each straight un- 
der its corresponding term, observing to carry 1 for every 12, from 
the inches to the feet. 

In like manner, multiply all the multiplicand by the inches and 
parts of the multiplier, and set the result of each term one place 
removed to the right hand of those in the multiplicand ; omitting, 
however, what is below parts of inches, only carrying to these the 
proper number of units from the lowest denomination. 



m 



THE PRACTICAL MODEL CALCULATOR. 



Or, instead of multiplying by the inches, take such parts of the 
multiplicand as these are of a foot. 

Then add the two lines together, after the manner of Compound Ad- 
dition, carrying 1 to the feet for 12 inches, when these come to so many. 
Multiply 4 f . 7 inc. Multiply 14 f. 9 inc. 
by 6 4 by ^ 6 



27 
1 


6 

H 

INVOLUTION. 


59 

7 




^ 


29 


66 


4| 



IxvoLUTiox is the raising of Powers from any given number, as 
a root. 

A Power is a quantity produced by multiplying any given num- 
ber, called the Root, a certain number of times continually by 
itself. Thus, 2 = 2 is the root, or first power of 2. 

2x2= 4 is the 2d power, or square of 2. 
2x2x2= 8 is the 3d power, or cube of 2. 
2 X 2 X 2 X 2 = 16 is the 4th power of 2, &c. 
And in this manner may be calculated the following Table of the 
first nine powers of the first nine numbers. 







lABLE OF 


IHE FIRST NINE 


POWERS 


OF NUMBERS. 


1st 


2d. 


3(1. 


4th. 


5tli. 


eth. 


7tli. 


8th. 


9th. 


1 


1 


1 


1 


1 


1 


1 


1 


1 


2 


4 


8 


16 


32 


64 


128 


256 


512 


3 


9 


27 


81 


243 


729 


2187 


6561 


19683 


4 


16 


64 


256 


1024 


4096 


16384 


65586 


262144 


5 


25 


125 


625 


3125 


15625 


78125 


390625 


1953125 


6 


36 


216 


1296 


7776 


46656 


279936 


1679616 


10077696 


7 


49 


343 


2401 


16807 


117619 


823543 


5764801 


40858607 


8 


64 


512 


4096 


32768 


262144 


2097152 


16777216 


134217728 


9 


81 


729 


6561 


59049 


581441 


4782969 


43046721 


387420489 



The Index or Exponent of a Power is the number denoting the 
height or degree of that power ; and it is 1 more than the number 
of multiplications used in producing the same. So 1 is the index 
or exponent of the 1st power or root, 2 of the 2d power or square, 
3 of the 3d power or cube, 4 of the 4th power, and so on. 

Powers, that are to be raised, are usually denoted by placing the 
index above the root or first power. 

So 2^ = 4, is the 2d power of 2. 
2' = 8, is the 3d power of 2. 
2^ = 16, is the 4th power of 2. 
540^ is the 4th power of 540 = 85030560000. 



EVOLUTION. 29 

When two or more powers are multiplied together, their product 
will be that power whose index is the sum of the exponents of the 
factors or powers multiplied. Or, the multiplication of the powers 
answers to the addition of the indices. Thus, in the following 



-owers of 2. 








1st. 2d. 3d. 


4th. 5th. 6th. 7th. 8th. 


9th. 


10th. 


2 4 8 


16 32 64 128 256 


512 


1024 


or, 2^ 2^ 2^ 


24 2^ 2^ 2'' 2^ 


2^ 


210 


Here, 4 X 


4 = 16, and 2 + 2 = 


4 its 


index ; 



and 8 X 16 = 128, and 3 + 4 = 7 its index ; 
also 16 X 64 = 1024, and 4 + 6 = 10 its index. 

The 2d power of 45 is 2025. 

The square of 446 is 17-3056. 

The 3d power of 3-5 is 42-875. 

The 5th power of -029 is -000000020511149. 

The square of § is f . 

The 3d power of | is |ff. 

The 4th power of f is ^. 

EVOLTJTIOIf. 

Evolution, or the reverse of Involution, is the extracting or 
finding the roots of any given powers. 

The root of any number, or power, is such a number as, being 
multiplied into itself a certain number of times, will produce that 
power. Thus, 2 is the square root or 2d root of 4, because 2^ = 

2 X 2 = 4 ; and 3 is the cube root or 3d root of 27, because 3^ = 

3 X 3 X 3 = 27. 

Any power of a given number or root may be found exactly, 
namely, by multiplying the number continually into itself. But 
there are many numbers of which a proposed root can never be 
exactly found. Yet, by means of decimals we may approximate 
or approach towards the root to any degree of exactness. 

These roots, which only approximate, are called Surd roots ; but 
those which can be found quite exact, are called Rational roots. 
Thus, the square root of 3 is a surd root ; but the square root of 

4 is a rational root, being equal to 2 : also, the cube root of 8 is 
rational, being equal to 2 ; but the cube root of 9 is surd, or 
irrational. 

Roots are sometimes denoted by writing the character s/ before 
the power, with the index of the root against it. Thus, the third 
root of 20 is expressed by -^20 ; and the square root or 2d root 
of it is \/20, the index 2 being always omitted w^hen the square 
root is designed. 

When the power is expressed by several numbers, with the sign 
+ or — between them, a line is drawn from the top of the sign over 
all the parts of it ; thus, the third root of 45 — 12 is a^45 — 12, 
or thus, ^(45 — 12), enclosing the numbers in parentheses. 
c2 



30 THE PRACTICAL MODEL CALCULATOR. 

But all roots are now often designed like powers, with fractional 
indices : thus, the square root of 8 is 8^, the cube root of 25 is 25 , 
and the 4th root of 45 - 18 is 45 - 18l^, or, (45 - 18)1 

TO EXTRACT THE SQUARE ROOT. 

Rule. — Divide the given number into periods of two figures 
each, by setting a point over the place of units, another over the 
place of hundreds, and so on, over every second figure, both to the 
left hand in integers, and to the right in decimals. 

Find the greatest square in the first period on the left hand, and 
set its root on the right hand of the given number, after the man- 
ner of a quotient figure in Division. 

Subtract the square thus found from the said period, and to the 
remainder annex the two figures of the next following period for a 
dividend. 

Double the root above mentioned for a divisor, and find how 
often it is contained in the said dividend, exclusive of its right-hand 
figure ; and set that quotient figure both in the quotient and divisor. 

Multiply the whole augmented divisor by this last quotient figure, 
and subtract the product from the said dividend, bringing down to 
the next period of the given number, for a new dividend. 

Repeat the same process over again, namely, find another new 
divisor, by doubling all the figures now found in the root ; from 
which, and the last dividend, find the next figure of the root as 
before, and so on through all the periods to the last. 

The best way of doubling the root to form the new divisor is by 
adding the last figure always to the last divisor, as appears in the 
following examples. Also, after the figures belonging to the given 
number are all exhausted, the operation may be continued into 
decimals at pleasure, by adding any number of periods of ciphers, 
two in each period. 

To find the square root of 29506624. 

29506624(5432 the root. 
25 



104 
4 


450 
416 


1083 
3 


3466 
3249 


10862 
2 


21724 
21724 



When the root is to he extracted to many places of figures, the work 

may he eonsiderahly shortened, thus : 

Having proceeded in the extraction after the common method till 

there be found half the required number of figures in the root, or 

one figure more j then, for the rest, divide the last remainder by 



TO EXTRACT THE SQUARE ROOT. 31 

its corresponding divisor, after the manner of the third contraction 
in Division of Decimals ; thus, 

To find the root of 2 to nine places of figures. 
2(1-4142 



24 100 
4 96 

281 400 
1 281 


2824 
4 


11900 
11296 


2828^ 


I 60400 

I 56564. 



28284) 3836(1356 
1008 
160 
19 
2 
1-41421356 the root required. 
The square root of -000729 is -02T. 
The square root of 3 is 1-T32050. 
The square root of 5 is 2-236068. 
The square root of 6 is 2-449489. 

RULES FOR THE SQUARE ROOTS OF COMMON FRACTIONS AND MIXED 

NUMBERS. 

First, prepare all common fractions by reducing them to their 
least terms, both for this and all other roots. Then, 

1. Take the root of the numerator and of the denominator for 
the respective terms of the root required. And this is the best 
way if the denominator be a complete power ; but if it be not, then, 

2. Multiply the numerator and denominator together ; take the 
root of the product : this root being made the numerator to the 
denominator of the given fraction, or made the denominator to the 
numerator of it, will form the fractional root required. 

. a y/a \/ah a 

And this rule will serve whether the root be finite or infinite. 

3. Or reduce the common fraction to a decimal, and extract its root. 

4. Mixed numbers may be either reduced to improper fractions, 
and extracted by the first or second rule ; or the common fraction 
may be reduced to a decimal, then joined to the integer, and the 
root of the whole extracted. 

The root of ff is f. 

The root of {-^ is f . 

The root of ^^ is 0-866025. 

The root of ^ is 0-645497. 

The root of 17| is 4-168333. 



32 



THE PRACTICAL MODEL CALCULATOR. 



By means of the square root, also, may readily be found the 4th 
root, or the 8th root, or the 16th root, &c. ; that is, the root of any 
power whose index is some power of the number 2 ; namely, by 
extracting so often the square root as is denoted by that power 
of 2 ; that is, two extractions for the 4th root, three for the 8th 
root, and so on. 

So, to find the 4th root of the number 21035-8, extract the 
square root twdce as follows : 



24 
4 


21035-8000 ( 145-037237 ( 12-0431407, the 4th root. 
1 1 

110 22 45 
96 2 44 


285 
5 


1435 2404 
1425 4 


10372 
9616 


290( 


)3 
6 


108000 24083 
87009 6 

20991(7237 
687 
107 
20 


75637 
72249 

3388(1407 
980 
17 



TO EXTRACT THE CUBE ROOT. 

1. Divide the page into three columns (i), (ii), (iii), in order, 
from left to right, so that the breadth of the columns may increase 
in the same order. In column (iii) write the given number, and 
divide it into periods of three figures each, by putting a point over 
the place of units, and also over every third figure, from thence to 
the left in whole numbers, and to the right in decimals. 

2. Find the nearest less cube number to the first or left-hand 
period ; set its root in column (iii), separating it from the right 
of the given number by a curve line, and also in column (i) ; then 
multiply the number in (i) by the root figure, thus giving the square 
of the first root figure, and write the result in (ii) ; multiply the 
number in (ii) by the root figure, thus giving the cube of the first 
root figure, and write the result below the first or left-hand period 
in (ill) ; subtract it therefrom, and annex the next period to the 
remainder for a dividend. 

3. In (i) write the root figure below the former, and multiply 
the sum of these by the root figure ; place the product in (ii), and 
add the two numbers together for a trial divisor. Again, write the 
root figure in (i), and add it to the former sum. 

4. With the number in (ii) as a trial divisor of the dividend, 
omitting the two figures to the right of it, find the next figure of 
the root, and annex it to the former, and also to the number in (i). 
Multiply the number now in (i) by the new figure of the root, and 
write the product as it arises in (ii), but extended two places of 
figures more to the right, and the sum of these two numbers will 
be the corrected divisor ; then multiply the corrected divisor by the 



TO EXTRACT THE CUBE ROOT. 



33 



last root figure, placing the product as it arises below tlie dividend ; 
subtract it therefrom, annex another period, and proceed precisely 
as described in (3), for correcting the columns (i) and (ii). Then 
with the new trial divisor in (ii), and the new dividend in (iii), 
proceed as before. 

When the trial divisor is not contained in the dividend, after two 
figures are omitted on the right, the next root figure is 0, and there- 
fore one cipher must be annexed to the number in (i) ; two ciphers 
to the number in (ii) ; and another period to the dividend in (in). 
When the root is interminable, we may contract the work very 
considerably, after obtaining a few figures in the decimal part of 
the root, if we omit to annex another period to the remainder in 
(in) ; cut ofi" one figure from the right of (ii), and two figures from 
(i), which will evidently have the efiect of cutting ofi" three figures 
from each column ; and then work with the numbers on the left, as 
in contracted multiplication and division of decimals. 
Find the cube root of 21035*8 to ten places of decimals. 

(I) (II) . . (Ill) 

2 4 21035-8 ( 27-60491055944 

2_ _8_ J 

4~ 12.. 13035 

2 46 9 11683 



67 

_7 
74 

7 



816 

6 

8 22 
6 



8 28 04 
4 

8 28 08 
4 

l-8i28|12 



1669 
518 
2187 . . 
4896 
223596 
4932 
228528 .... 
331216 
2285611216 
331232 



228594244 
7453 



228601697 
7453 



2286091511 

8J3 

228 6 092 314 
83 



1352800 
1341576 

11224 

9142444864 

2081555136 

2057415281 

24139855 

22860923 

1278932 

1143046 

135886 

114305 

21581 

20575 

1006 

914 

92 

• 91 



2|218|6|0|9|3|2 1 

Kequired the cube roots of the following numbers :- 



48228544, 46656, and 15069223. 
64481-201, and 28991029248. 
12821119155125, and -000076765625. 
ififl, and 16. 
911 and 7f. 

3 



364, 36, and 247. 

40-1, and 3072. 

23405, and -0425. 

ft, and 2-519842. 

4-5, and 1-98802366. 



34 THE PRACTICAL MODEL CALCULATOR. 

TO EXTRACT ANY ROOT WHATEVER. 

Let N be the given power or number, n the index of the power, 
A the assumed power, r its root, R the required root 6f N. 

Then, as the sum of w + 1 times A and n — 1 times N, is to 
the sum oi n -{• 1 times N and n — 1 times A, so is the assumed 
root r, to the required root E,. 

Or, as half the said sum of ?z + 1 times A and n — 1 times N, 
is to the difference between the given and assumed powers, so is the 
assumed root r, to the difference between the true and assumed 
roots ; which difference, added or subtracted, as the case requires, 
gives the true root nearly. 

That is, (?^ + 1) . A + (?z - 1) . N : (n + 1) . N + (tz - 1) • A : : r : R. 
Or, {n + 1) . JA + (w - 1) . IN : A 02 :N' : ; r : R c/2 r. 

And the operation may be repeated as often as we please, by 
using always the last found root for the assumed root, and its ?ith 
power for the assumed power A. 

To extract the 5th root o/ 21035-8. 

Here it appears that the 5th root is between 7*3 and 7*4. Taking 
7*3, its 5th power is 20730-71593. Hence then we have, 

N = 21035-8; r = 7-3; 7i = 5; J . {n + 1) = 3; }.(?i - 1) = 2. 

A = 20730-716 
K-A = 305-084 



A = 20730-716 N = 21035-8 



3 A = 62192.148 42071-6 
2N = 42071-6 

As 104263-7 : 305-084 : : 7-3 : -0213605 
7-3 



915252 
2135588 



104263-7) 2227-1132 ( -0213605, the difference. 
14184 7-3 = r add 

3758 

630 7-321360 = R, the root, true to 
5 the last figure. 

The 6th root of 21035.8 is 5-254037. 

The 6th root of 2 is 1-122462. 

The 7th root of 21035-8 is 4-145392. 

The 7th root of 2 is 1-104089. 

The 9th root of 2 is 1-080059. 

OF RATIOS, PROPORTIONS, AND PROGRESSIONS. 

Numbers are compared to each other in two different ways : the 
one comparison considers the difference of the two numbers, and 
is named Arithmetical Relation, and the difference sometimes 
Arithmetical Ratio : the other considers their quotient, and is called 



ARITHMETICAL PROPORTION AND PROGRESSION. 35 



Geometrical E-elation, and the quotient the Geometrical Ratio. So, 
of these two numbers 6 and 3, the difference or arithmetical ratio 
is 6 — 3 or 3 ; but the geometrical ratio is f or 2. 

There must be two numbers to form a comparison : the number 
which is compared, being placed first, is called the Antecedent ; 
and that to which it is compared the Consequent. So, in the 
two numbers above, 6 is the antecedent, and 3 is the consequent. 

If two or more couplets of numbers have equal ratios, or equal 
differences, the equality is named Proportion, and the terms of the 
ratios Proportionals. So, the two couplets, 4, 2 and 8, 6 are arith- 
metical proportionals, because 4 — 2 = 8 — 6 = 2; and the two cou- 
plets 4, 2 and 6, 3 are geometrical proportionals, because | = f = 2, 
the same ratio. 

To denote numbers as being geometrically proportional, a colon 
is set between the terms of each couplet to denote their ratio ; and 
a double colon, or else a mark of equality between the couplets or 
ratios. So, the four proportionals, 4, 2, 6, 3, are set thus, 4 : 2 : : 6 : 3, 
which means that 4 is to 2 as 6 is to 3 ; or thus, 4:2 = 6:3; or 
thus, 1 = 1, both which mean that the ratio of 4 to 2 is equal to 
the ratio of 6 to 3. 

Proportion is distinguished into Continued and Discontinued. 
When the difference or ratio of the consequent of one couplet and 
the antecedent of the next couplet is not the same as the common 
difference or ratio of the couplets, the proportion is discontinued. 
So, 4, 2, 8, 6 are in discontinued arithmetical proportion, because 
4-2 = 8-6 = 2, whereas, 2 - 8 = - 6; and 4, 2, 6, 3 are in 
discontinued geometrical proportion, because -| = f = 2, but f = |, 
which is not the same. 

But when the difference or ratio of every two succeeding terms is 
the same quantity, the proportion is said to be continued, and the num- 
bers themselves a series of continued proportionals, or a progression. 
So, 2, 4, 6, 8 form an arithmetical progression, because 4 — 2 = 6 — 
4 = 8 — 6 = 2, all the same common difference ; and 2, 4, 8, 16, a 
geometrical progression, because | = | = -V® = 2, all the same ratio. 

When the foUoAving terms of a Progression exceed each other, 
it is called an Ascending Progression or Series ; but if the terms 
decrease, it is a Descending one. 

So, 0, 1, 2, 3, 4, &c., is an ascending arithmetical progression, 

but 9, 7, 5, 3, 1, &c., is a descending arithmetical progression : 

Also, 1, 2, 4, 8, 16, &c., is an ascending geometrical progression, 

and 16, 8, 4, 2, 1, &c., is a descending geometrical progression. 
ARITHMETICAL PROPORTION AND PROGRESSION. 

The first and last terms of a Progression are called the Extremes ; 
and the other terms lying between them, the Means. 

The most useful part of arithmetical proportions is contained in 
the following theorems : 

Theorem 1. — If four quantities be in arithmetical proportion, the 
sum of the two extremes will be equal to the sum of the two means. 

Thus, of the four 2, 4, 6, 8, here 2 + 8 = 4 + 6= 10. 



36 THE PRACTICAL MODEL CALCULATOR. 

Theorem 2. — In any continued arithmetical progression, tlie sum 
of the two extremes is equal to the sum of any two means that are 
equally distant from them, or equal to double the middle term when 
there is an uneven number of terms. 

Thus, in the terms 1, 3, 5, it is 1 + 5 = 3 + 3 = 6. 

And in the series 2, 4, 6, 8, 10, 12, 14, it is 2 + 14 = 4 + 12 = 
6 + 10 = 8 + 8 = 16. 

Theorem 3. — The difference between the extreme terms of an 
arithmetical progression, is equal to the common difference of the 
series multiplied by one less than the number of the terms. 

So, of the ten terms, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, the com- 
mon difference is 2, and one less than the number of terms 9 ; then 
the difference of the extremes is 20 — 2 = 18, and 2 x 9 = 18 also. 

Consequently, the greatest term is equal to the least term added 
to the product of the common difference multiplied by 1 less than 
the number of terms. 

Theorem 4. — The sum of all the terms of any arithmetical pro- 
gression is equal to the sum of the two extremes multiplied by the 
number of terms, and divided by 2 ; or the sum of the two extremes 
multiplied by the number of the terms gives double the sum of all 
the terms in the series. 

This is made evident by setting the terms of the series in an 
inverted order under the same series in a direct order, and adding 
the corresponding terms together in that order. Thus, 

in the series, 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 ; 

inverted, 15 , 13 , 11 , 9 , 7 , 5 , 3 , 1 ; 

the sums are, 16 -f- 16 + 16 + 16 + 16 -f 16 + 16 + 16, 
which must be double the sum of the single series, and is equal to 
the sum of the extremes repeated so often as are the number of 
the terms. 

From these theorems may readily be found any one of these five 
parts ; the two extremes, the number of terms, the common differ- 
ence, and the sum of all the terms, when any three of them are 
given, as in the following Problems : 

problem I. 
Criven the extreriie§ and tlie number of temUy to find tlie sum of all 

the terms. 
Rule. — Add the extremes together, multiply the sum by the 
number of terms, and divide by 2. 

The extremes being 3 and 19, and the number of terms 9 ; 
required the sum of the terms ? 
19 

2)198 

99 = the sum. 



ARITHMETICAL PROPORTION AND PROGRESSION. 37 

The strokes a clock strikes in one whole revolution of the index, 
or in 12 hours, is 78. 

PROBLEM II. 

Griven the extremes, and the number of terms ; to find the common 

differe7ice. 

Rule. — Subtract the less extreme from the greater, and divide 
the remainder by 1 less than the number of terms, for the common 
difference. 

The extremes being 3 and 19, and the number of terms 9 ; re- 
quired the common difference ? 
19 

J. ^ 19-3 16 _ 
8)16 ^^' T=T=T=^* 
_2 

If the extremes be 10 and 70, and the number of terms 21 ; what 
is the common difference, and the sum of the series ? 

The com. diff. is 3, and the sura is 840. 

PROBLEM in. 

Given one of the extremes, the common difference, and the number 
of terms; to find the other extreme, and the sum of the series. 
Rule. — Multiply the common difference by 1 less than the num- 
ber of terms, and the product will be the difference of the extremes : 
therefore add the product to the less extreme, to give the greater ; 
or subtract it from the greater, to give the less. 

Given the least term 3, the common difference 2, of an arith- 
metical series of 9 terms ', to find the greatest term, and the sum 
of the series ? 



16 

3 

19 the greatest term. 

3 the least. 
"22 sum. 
9 number of terms. 
2)198 

99 the sum of the series. 

If the greatest term be 70, the common difference 3, and the 
number of terms 21 ; what is the least term and the sum of the 
series ? The least term is 10, and the sum is 840. 

PROBLEM IV. 

To find an arithmetical mean proijortional between two given terms. 

Rule. — Add the two given extremes or terms together, and take 

half their sum for the arithmetical mean requii'ed. Or, subtract 



38 THE PRACTICAL MODEL CALCULATOR. 

tlie less extreme from the greater, and half the remainder will be 
the common difference ; "which, being added to the less extreme, or 
subtracted from the greater, will give the mean required. 

To find an arithmetical mean between the two numbers 4 and 14. 



re, 14 


Or, 14 Or, 14 


4 


4 5 


2)18 


2)10 9 


9 


5 the com. dif. 




4 the less extreme. 




9 



So that 9 is the mean required by both methods. 

PROBLEM Y. 

To find tivo arithmetical means between two given extremes. 

Rule. — Subtract the less extreme from the greater, and divide 
the difference by 3, so will the quotient be the common difference ; 
which, being continually added to the less extreme, or taken from 
the greater, gives the means. 

To find two arithmetical means between 2 and 8. 

Here 8 

^ Then 2 + 2 = 4 the one mean, 
^ ) ^ and 4 + 2 = 6 the other mean, 

com. dif. 2 

problem VI. 
To find any number of arithmetical means between two given terms 

or extremes. 
Rule. — Subtract the less extreme from the greater, and divide 
the difference by 1 more than the number of means required to be 
found, which will give the common difference ; then this being 
added continually to the least term, or subtracted from the greatest, 
will give the mean terms required. 

To find five arithmetical means between 2 and 14. 
Here 14 

o 

Then, by adding this com. dif. continually, 

6 )22 the means are found, 4, 6, 8, 10, 12. 
com. dif. 2 

GEOMETRICAL PROPORTION AND PROGRESSION. 

The most useful part of Geometrical Proportion is contained in 
the following theorems : 

Theorem 1. — If four quantities be in geometrical proportion, 
the product of the two extremes will be equal to the product of the 
two means. 

Thus, in the four 2, 4, 3, 6 it is 2 x 6 = 3 X 4 = 12. 

And hence, if the product of the two means be divided by one 
of the extremes, the quotient will give the other extreme. So, of 



GEOMETRICAL PROPORTION AND PROGRESSION. 89 

the above numbers, the product of the means 12 -^ 2 = 6 the one 
extreme, and 12 -j- 6 = 2 the other extreme ; and this is the 
foundation and reason of the practice in the Rule of Three. 

Theorem 2. — In any continued geometrical progression, the pro- 
duct of the two extremes is equal to the product of any two means 
that are equally distant from them, or equal to the square of the 
middle term when there is an uneven number of terms. 

Thus, in the terms 2, 4, 8, it is 2 x 8 = 4 X 4 = 16. 
And in the series 2, 4, 8, 16, 32, 64, 128, 

it is 2 X 128 = 4 X 64 = 8 X 32 = 16 X 16 = 256. 

Theorem 3. — The quotient of the extreme terms of a geome- 
trical progression is equal to the common ratio of the series raised 
to the power denoted by one less than the number of the terms. 

So, of the ten terms 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 
the common ratio is 2, one less than the number of terms 9 ; then 

1024 
the quotient of the extremes is — ^ — ~ ^^^j ^^^ ^^ ~ ^^^ ^^^^• 

Consequently, the greatest term is equal to the least term multi- 
plied by the said power of the ratio whose index is one less than 
the number of terms. 

Theorem 4. — The sum of all the terms of any geometrical pro- 
gression is found by adding the greatest term to the difference of 
the extremes divided by one less than the ratio. 

So, the sum 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, (whose ratio 
1024 - 2 
is 2,) is 1024 + ^_^ = 1024 -f 1022 = 2046. 

The foregoing, and several other properties of geometrical pro- 
portion, are demonstrated more at large in Byrne's Doctrine of Pro- 
portion. A few examples may here be added to the theorems just 
delivered, with some problems concerning mean proportionals. 

The least of ten terms in geometrical progression being 1, and 

the ratio 2, what is the greatest term, and the sum of all the terms ? 

The greatest term is 512, and the sum 1023. 

problem I. 
To find one geometrical 7nean proportional between any two numbers. 

Rule. — Multiply the two numbers together, and extract the square 
root of the product, which will give the mean proportional sought. 

Or, divide the greater term by the less, and extract the square 
root of the quotient, which will give the common ratio of the three 
terms : then multiply the less term by the ratio, or divide the 
greater term by it, either of these will give the middle term required. 

To find a geometrical mean between the two numbers 3 and 12. 

First ivay. Second ivay. 

12 3 ) 12 ( 4, its root, is 2, the ratio. 

36 ( 6 the mean. Then, 3x2 = 6 the mean. 

36 Or, 12 -f- 2 = 6 also. 



40 THE PRACTICAL MODEL CALCULATOR. 

PROBLEM II. 

To find two geometrical mean ijroportionaU hetiveen any two numbers. 
Rule. — Divide the greater number by tlie less, and extract the 
cube root of the quotient, which -will give the common ratio of the 
terms. Then multiply the least given term by the ratio for the 
first mean, and this mean again by the ratio for the second mean ; 
or, divide the greater of the two given terms by the ratio for the 
greater mean, and divide this again by the ratio for the less mean. 
To find two geometrical mean proportionals between 3 and 24. 
Here, 3 ) 24 ( 8, its cube root, 2 is the ratio. 
Then, 3x2= 6, and 6 X 2 = 12, the two means. 
Or, 24 -^ 2 = 12, and 12 -r- 2 = 6, the same. 

That is, the two means between 3 and 24, are 6 and 12. 

PROBLEM ni. 
To fund any oiumher of geometrical mean proportionals between tivo 

numbers. 

Rule. — Divide the greater number by the less, and extract such 
root of the quotient whose index is one more than the number of 
means required, that is, the 2d root for 1 mean, the 3d root for 
2 means, the 4th root for 3 means, and so on ; and that root will 
be the common ratio of all the terms. Then with the ratio multi- 
ply continually from the first term, or divide continually from the 
last or greatest term. 

To find four geometrical mean proportionals between 3 and 96. 

Here, 3 ) 96 ( 32, the 5th root of which is 2, the ratio. 
Then, 3x2= 6, and 6 x 2=12, and 12 x 2=24, and 24 X 2=48. 
Or, 96 -^ 2=48, and 48 -f-2=24, and 24-^2=12, and 12--2= 6. 
That is, 6, 12, 24, 48 are the four means between 3 and 96. 

OF MUSICAL PROPORTION. 

There is also a third kind of proportion, called Musical, which, 
being but of little or no common use, a very short account of it may 
here suffice. 

Musical proportion is when, of three numbers, the first has the 
same proportion to the third, as the difference between the first and 
second has to the difference between the second and third. 

As in these three, 6, 8, 12 ; 

where, 6 : 12 : : 8 - 6 : 12 - 8, 

that is, 6 : 12 : : 2 : 4. 
When four numbers are in Musical Proportion ; then the first 
has the same proportion to the fourth, as the difference between 
the first and second has to the difference between the third and 
fourth. 

As in these, 6, 8, 12, 18 ; 

where, 6 : 18 : : 8 - 6 : 18 - 12, 

that is, 6 : 18 : : 2 : 6. 



FELLOWSHIP. 41 

When numbers are in Musical Progression, their reciprocals are 
in Arithmetical Progression ; and the converse, that is, when num- 
bers are in Arithmetical Progression, their reciprocals are in Mu- 
sical Progression. 

So, in these Musicals 6, 8, 12, their reciprocals f, |, ^, are in 
arithmetical progression ; for i + ^ = o = i ; and i + | = f = i ; 
that is, the sum of the extremes is equal to double the mean, which 
is the property of arithmeticals. 

FELLOWSHIP, OR PARTNERSHIP. 

Fellowship is a rule by which any sum or quantity may be 
divided into any number of parts, which shall be in any given pro- 
portion to one another. 

By this rule are adjusted the gains, or losses, or charges of part- 
ners in company ; or the effects of bankrupts, or legacies in case of 
a deficiency of assets or effects ; or the shares of prizes, or the 
numbers of men to form certain detachments ; or the division of 
waste lands among a number of proprietors. 

Pellowship is either Single or Double. It is Single, when the 
shares or portions are to be proportional each to one single given 
number only ; as when the stocks of partners are all employed for 
the same time : and Double, when each portion is to be proportional 
to two or more numbers ; as when the stocks of partners are em- 
ployed for different times. 

SINGLE fellowship. 

General Rule. — Add together the numbers that denote the 
proportion of the shares. Then, 

As the sum of the said proportional numbers 
Is to the whole sum to be parted or divided. 
So is each several proportional number 
To the corresponding share or part. 
Or, As the whole stock is to the whole gain or loss, 

So is each man's particular stock to his particular share of 
the gain or loss. 
To prove the work. — Add all the shares or parts together, and 
the sum will be equal to the whole number to be shared, when the 
work is right. 

To divide the number 240 into three such parts, as shall be in 
proportion to each other as the three numbers, 1, 2, and 3. 
Here 1 + 2 + 3 = 6 the sum of the proportional numbers. 
Then, as 6 : 240 : : 1 : 40 the 1st part, 
and, as 6 : 240 : : 2 : 80 the 2d part, 
also as 6 : 240 : : 3 : 120 the 3d part. 
Sum of all 240, the proof. 

Three persons, A, B, C, freighted a ship with 340 tuns of wine ; 
of which, A loaded 110 tuns, B 97, and C the rest : in a storm, the 
d2 



ence. 


as 


340 : 


85 


or, 


as 


4: 


1 


and, 


as 


4: 


1 


also, 


as 


4: 


1 



42 THE PRACTICAL MODEL CALCULATOR. 

seamen were obliged to throw overboard 85 tuns ; how much must 
each person sustain of the loss ? 

Here, 110 + 97 = 20T tuns, loaded by A and B ; 

theref., 340 - 20T = 133 tuns, loaded by C. 
~- - 110 

110 : 271 tuns = A's loss ; 
97 : 24-1 tuns = B's loss ; 
133 : 33| tuns = C's loss. 
Sum 85 tuns, the proof. 

DOUBLE FELLOWSHIP. 

Double Fellowship, as has been said, is concerned in cases 
in which the stocks of partners are employed or continued for dif- 
ferent times. 

Rule. — Multiply each person's stock by the time of its continu- 
ance ; then divide the quantity, as in Single Eellowship, into shares 
in proportion to these products, by saying : 

As the total sum of all the said products 

Is to the whole gain or loss, or quantity to be parted, 

So is each particular product 

To the corresponding share of the gain or loss. 

SIMPLE INTEREST. 

Interest is the premium or sum allowed for the loan, or for- 
bearance of money. 

The money lent, or forborne, is called the Principal. 

The sum of the principal and its interest, added together, is 
called the Amount. 

Interest is allowed at so much per cent, per annum, which pre- 
mium per cent, per annum, or interest of a ^100 for a year, is 
called the Eate of Interest. So, 

When interest is at 3 per cent, the rate is 3 ; 

4 per cent 4; 

5 per cent 5; 

6 per cent 6. 

Interest is of two sorts : Simple and Compound. 

Simple Interest is that which is allowed for the principal lent or 
forborne only, for the whole time of forbearance. 

As the interest of any sum, for any time, is directly proportional 
to the principal sum, and also to the time of continuance ; hence 
arises the following general rule of calculation. 

GrENERAL RuLE. — As $100 is to the rate of interest, so is any 
given principal to its interest for one year. And again. 

As one year is to any given time, so is the interest for a year just 
found to the interest of the given sum for that time. 

Otlienvise. — Take the interest of one dollar for a year, which, 
multiply by the given principal, and this product again by the time 



POSITION. 43 

of loan or forbearance, in years and parts, for the interest of the 
proposed sum for that time. 

When there are certain parts or years in the time, as quarters, 
or months, or days, they may be worked for either by taking the 
aliquot, or like parts of the interest of a year, or by the Rule of 
Three, in the usual way. Also, to divide by 100, is done by only 
pointing off two figures for decimals. 

COMPOUND INTEREST. 

Compound Interest, called also Interest upon Interest, is that 
which arises from the principal and interest, taken together, as it 
becomes due at the end of each stated time of payment. 

KuLES. — 1. Find the amount of the given principal, for the time 
of the first payment, by Simple Interest. Then consider this 
amount as a new principal for the second payment, whose amount 
calculate as before ; and so on, through all the payments to the last, 
always accounting the last amount as a new principal for the next 
payment. The reason of which is evident from the definition of 
Compound Interest. Or else, 

2. Find the amount of one dollar for the time of the first pay- 
ment, and raise or involve it to the power whose index is denoted 
by the number of payments. Then that power multiplied by the 
given principal will produce the whole amount. From which the 
said principal being subtracted, leaves the Compound Interest of 
the same ; as is evident from the first rule. 

POSITION. 

Position is a method of performing certain questions which can- 
not be resolved by the common direct rules. It is sometimes called 
False Position, or False Supposition, because it makes a supposi- 
tion of false numbers to work with, the same as if they were the 
true ones, and by their means discovers the true numbers sought. 
It is sometimes also called Trial and Error, because it proceeds 
by trials of false numbers, and thence finds out the true ones by a 
comparison of the errors. 

Position is either Single or Double. 

SINGLE POSITION. 

Single Position is that by which a question is resolved by means 
of one supposition only. 

Questions which have their results proportional to their supposi- 
tions belong to Single Position ; such as those w^iich require the 
multiplication or division of the number sought by any proposed 
number ; or, when it is to be increased or diminished by itself, or 
any parts of itself, a certain proposed number of times. 

Rule. — Take or assume any number for that required, and per- 
form the same operations with it as are described or performed in 
the question. 

Then say, as the result of the said operation is to the position 



44 THE PRACTICAL MODEL CALCULATOR. 

or number assumed, so is tlie result in the question to the number 
sought. 

A person, after spending J and J of his money, has yet remain- 
ino; S60, what had he at first? 



Suppose he had at first ^120 Proof. 

Now i of 120 is 40 J of 144 is 48 

1 of it is JO i of 144 is _36 

their sum is 70 their sum 84 

which taken from 120 taken from 144 

leaves 50 leaves 60 as per question. 

Then, 50 : 120 : : 60 : 144. 

"What number is that, which multiplied by 7, and the product 
divided by 6, the quotient may be 14 ? 12. 

PERMUTATIONS AND COMBINATIONS. 

The Permutations of any number of quantities signify the changes 
which these quantities may undergo with respect to their order. 

Thus, if we take the quantities a, h, c; then, a h c, a c h, b a c, 
h c a, c a b, c h a, are the permutations of these three quantities 
taken all together; a b, a c, b a, b c, c a, c b, are the permutations 
of these quantities taken two and two; a, b, c, are the permutation 
of these quantities taken singly, or one and one, &c. 

The number of the permutations of the eight letters, a, 6, c, d, 
e, /, (/, 7i, is 40320 ; becomes, 

1.2.3.4.5.6.7.8= 40320. 

DOUBLE POSITION. 

Double Position is the method of resolving certain questions 
by means of two suppositions of false numbers. 

To the Double Rule of Position belong such questions as have 
their results not proportional to their positions : such are those, in 
which the numbers sought, or their parts, or their multiples, are 
increased or diminished by some given absolute number, which is 
no known part of the number sought. 

Take or assume any two convenient numbers, and proceed with 
each of them separately, according to the conditions of the ques- 
tion, as in Single Position ; and find how much each result is dif- 
ferent from the result mentioned in the question, noting also 
whether the results are too great or too little. 

Then multiply each of the said errors by the contrary supposi- 
tion, namely, the first position by the second error, and the second 
position by the first error. 

If the errors are alike, divide the difi*erence of the products by 
the difference of the errors, and the quotient will be the answer. 

But if the errors are unlike, divide the sum of the products by 
the sum of the errors, for the answer. 

The errors are said to be alike, when they are either both too 
great, or both too little ; and unlike, when one is too great and the 
other too little. 



MENSURATION OF SUPERFICIES. 45 

"VYliat number is that, which, being multiplied bj 6, the product 
increased by 18, and the sum divided by 9, the quotient shall be 20. 
Suppose the two numbers, 18 and 30. Then 



First position. . 
18 


Second position. 
30 


Proof. 

27 


6 mult. 


6 


6 


108 


180 


162 


18 add. 


18 


18 


9)126 


9)198 


9)180 


14 results. 


22 


20 


20 true res. 


20 




+ 6 errors unlike. 


— 2 




2d pos. 30 mult. 


18 1st pos. 




Errors {^^«« 


36 




Sum 8 ) 216 sum of products 






27 answer sought. 







Find, by trial, two numbers, as near the true number as possible, 
and operate with them as in the question ; marking the errors 
which arise from each of them. 

Multiply the difference of the two numbers, found by trial, by 
the least error, and divide the product by the difference of the 
errors, when they are alike, but by their sum when they are unlike. 

Add the quotient, last found, to the number belonging to the 
least error, when that number is too little, but subtract it when too 
great, and the result will give the true quantity sought. 

MENSURATION OF STJPEREICIES. 

The area of any figure is the measure of its surface, or the space 
contained within the bounds of that surface, without any regard to 
thickness. 

A square whose side is one inch, one foot, or one yard, &c. is 
called the measuring unit, and the area or content of any figure is 
computed by the number of those squares contained in that figure. 

To find the area of a imrallelogram ; whether it he a square, a 
rectangle, a rhombus, or a rhomhoides. — Multiply the length by the 
perpendicular height, and the product will be the area. 

The perpendicular height of the parallelogram is equal to the 
area divided by the base. 



Required the area of the square ABCD whose 
side is 5 feet 9 inches. 

Here 5 ft. 9 in, = 5-75 : aiid'd^l^ = 5'75 X 
5-75 = 33-0625 /ee^ = 33/e. in. ^ pa. = area 
required. 



46 



THE PRACTICAL MODEL CALCULATOR. 



Required the area of tlie rectangle 
ABCD, whose length AB is 13-75 chains, 
and breadth BC 9-5 chains. 

Here 13-75 X 9-5 = 130-625; and 

-^q" = 13-0625 ac. = 13 ac. ro. 10 
po. = area required. 

Required the area of the rhombus 
ABCD, whose length AB is 12 feet 6 
inches, and its height DE 9 feet 3 inches. 
= 12*5, and 9 fe. 3 in. 




Sere 12 fe. 6 in. 
= 9-25. 

Wlience, 12-5 X 9-25 = 115-625 fe. = 
115 fe. 7 in. 6 ]9a. = area required. 

What is the area of the rhom- 
boides ABCD, whose length AB is 
10-52 chains, and height DE 7-63 
chains. 

Here 10-52 X 7-63 = 80-2676 ; 

80-2676 
and — T7^ — = 8-02676 acres = 8 ae. 




10 




7'o. 4po. area required. 

To find the area of a triangle. — Multiply the base by the per- 
pendicular height, and half the product will be the area. 

The perpendicular height of the triangle is equal to twice the 
area divided by the base. 

Required the area of the triangle ABC, 
whose base AB is 10 feet 9 inches, and 
height DC 7 feet 3 inches. 

Here 10 fe. 9 in. = 10-75, and 7 fe. 3 in. 
= 7-25. 

Whe7ice, 10-75 X 7-25 = 77-9375, and 
77-9375 
— 2 = 38-96875 feet = 38 fe. 11 in. 

7 J pa. = area required. 

To find the area of a triangle whose three sides only are given. — 
From half the sum of the three sides subtract each side severally. 
Multiply the half sum and the three remainders continually toge- 
ther, and the square root of the product will be the area required- 
Required the area of the triangle ABC, 
whose three sides BC, CA, and AB are 
24, 36, and 48 chains respectively. 

^ 24 + 36 + 48 108 

Here ^ = -^ = 54 = 

J sum of the sides. 

Also, 54 - 24 = 2>0 first diff. ; 54 - 36 
= 18 second diff.; and 54 ■— 48 = 6 third diff. 




MENSURATION OF SUPERFICIES. 



47 



Whence, s/ 54: x 30 x 18 x 6 = n/ 174960 = 418*282 = area 
required. 

Any two sides of a right angled triangle being given to find 
the third side. — When the two legs are given to find the hypo- 
thenuse, add the square of one of the legs to the square of the 
other, and the square root of the sum will be equal to the hypo- 
thenuse. 

Wien the hypothenuse and one of the legs are given to find the 
other leg. — From the square of the hypothenuse take the square 
of the given leg, and the square root of the remainder will be equal 
to the othe^r leg. 

In the right angled triangle ABC, the c 

base AB is bQ, and the perpendicular BC 33, 
what is the hypothenuse ? 

Here bQ^ + 33^ = 3136 + 1089 = 4225, 
and v^4225 = Qb = hypothenuse AC. 

If the hypothenuse AC be 53, and the 
base AB 45, what is the perpendicular BC ? 

Here 53^ - 45^ = 2809 - 2025 = 784, and v/784 = 28 =- 
2?erpendicular BC. 

To find the area of a trapezium. — Multiply the diagonal by the 
sum of the two perpendiculars falling upon it from the opposite 
angles, and half the product will be the area. 

Required the area of the trapezium 
BAED, whose diagonal BE is 84, the 
perpendicular AC 21, and DF 28. 

Here 28 + 21 X 84 = 49 X 84= 




and 



4116 



=4116, 

2058 the area required. 




To find the area of a trapezoid, or a quadrangle, two of whose 
opposite sides are parallel. — Multiply the sum of the parallel sides 
by the perpendicular distance between them, and half the product 
will be the area. 

Required the area of the trapezoid ABCD, 
whose sides AB and DC are 321-51 and 
214-24, and perpendicular DE 171-16. 

Here 321-51 + 214-24 = 535-75 = sum 
of the parallel sides AB, DC. 

Whence, 535-75 X 171-16 (theperp. DE) = 
^1698-9700 
91698-9700, and ~ ^ = 45849-485 the area required. 

To find the area of a regular polygon. — Multiply half the peri- 
meter of the figure by the perpendicular falling from its centre 
upon one of the sides, and the product will be the area. 

The perimeter of any figure is the sum of all its sides. 




48 



THE PRACTICAL MODEL CALCULATOR. 



Required the area of the regular pentagon 

ABODE, whose side AB, or BC, &c., is 25 

feet, and the perpendicular OP 17*2 feet. 

25 X 5 
Here — o — = 62-5 = half perimeter, 



and 62-5 X lT-2 
required. 



1075 square feet = area 




To find the area of a regular polygon, when the side only is 
given. — Multiply the square of the side of the polygon by the 
number standing opposite to its name in the following table, and 
the product will be the area. 



No. of 
sides. 


Names. 


Mvdtipliers. 


No. of 
sides. 


Names. 


Multipliers. 


3 
4 

5 
6 

7 


Trigon or equil. A 
Tetragon or square 
Pentagon 
Hexagon 
Heptagon 


0-433013 
1-000000 
1-720477 
2-598076 
3-633912 


8 

9 

10 

11 

12 


Octagon 

Nonagon 

Decagon 

Undecagon 

Duodecagon 


4-828427 
6-181824 
7-694209 
9-365640 
11-196152 



The angle OBP, together with its tangent, for any polygon of not 
more than 12 sides, is shown in the following table : 



No. of 
sides. 


Names. 


Angle 
CEP. 


Tangents. 


3 
4 

5 
6 

7 
8 
9 

10 
11 
12 


Trigon 
Tetragon 

Pentagon 

Hexagon 

Heptagon 

Octagon 

Nonagon 

Decagon 

Undecagon 

Duodecagon 


30° 

45° 

54° 

60° 

64°f 

67°J 

70° 

72° 


-57735 = 1 ^3 
1-00000 = 1X1 

1-37638 = v^l + f v/o 

1-73205 = ^B 

2-07652 

2-41421 = 1 + v^2 

2-74747 


3-07768 = v'S + 2 v^6 

3-40568 

3-73205 = 2 + ^3 



Required the area of a pentagon whose side is 15. 

The number opposite pentagon in the table is 1*720477. 

m^ice 1-720477 x 15^ = 1-720477 x 225 = 387-107325 = 
area required. 

The diameter of a circle being given to find the circumference, 
or the ch^cumference being given to find the diameter. — Multiply 
the diameter by 3-1416, and the product will be the circumfer- 
ence, or 

Divide the circumference by 3-1416, and the quotient will be the 
diameter. 

As 7 is to 22, so is the diameter to the circumference ; or as 22 
is to 7, so is the circumference to the diameter. 

As 113 is to 355, so is the diameter to the circumference ; or, 
as 352 is to 115, so is the circumference to the diameter. 



MENSURATION OF SUPERFICIES. 49 

If the diameter of a circle be 17, what is the circumference ? 

Here 3-1416 X 17 = 53-4072 = circumference. 
If the circumference of a circle be 354, what is the diameter ? 
354-000 
Here q.^^-ip = 112-681 = diameter. 

To find the length of any arc of a circle. — When the chord of 
the arc and the versed sine of half the arc are given : 

To 15 times the square of the chord, add 33 times the square of 
the versed sine, and reserve the number. 

To the square of the chord, add 4 times the square of the versed sine, 
and the square root of the sum will be twice the chord of half the arc. 

Multiply twice the chord of half the arc by 10 times the square 
of the versed sine, divide the product by the reserved number, and 
add the quotient to twice the chord of half the arc : the sum will 
be the length of the arc very nearly. 

When the chord of the arc, and the chord of half the arc are 
given. — From the square of the chord of half the arc subtract the 
square of half the chord of the arc, the remainder will be the square 
of the versed sine : then proceed as above. 

When the diameter and the versed sine of half the arc are given : 

From 60 times the diameter subtract 27 times the versed sine, 
and reserve the number. 

Multiply the diameter by the versed sine, and the square root 
of the product will be the chord of half the arc. 

Multiply twice the chord of half the arc by 10 times the versed 
sine, divide the product by the reserved number, and add the quo- 
tient to twice the chord of half the arc ; the sum will be the length 
of the arc very nearly. 

When the diameter and chord of the arc are given, the versed 
sine may be found thus : From the square of the diameter subtract 
the square of the chord, and extract the square root of the re- 
mainder. Subtract this root from the diameter, and half the re- 
mainder will give the versed sine of half the arc. 

The square of the chord of half the arc being divided by the 
diameter will give the versed sine, or being divided by the versed 
sine will give the diameter. 

The length of the arc may also be found by multiplying together the 
number of degrees it contains, the radius and the number -01745329. 

Or, as 180 is to the number of degrees in the arc, so is 3-1416 
times the radius, to the length of the arc. 

Or, as 3 is to the number of degrees in the arc, so is -05236 times 
the radius to the length of the arc. 

If the chord DE be 48, and the versed sine 
CB 18, what is the length of the arc ? 
Here 48^ x 15 = 34560 
182 X 33 = 10692 

45252 reserved numher. 




50 THE PRACTICAL MODEL CALCULATOR. 

48^ = 2304 = the square of tlie cJiord. 
18^ X 4 = 1296 = 4 times the square of the versed sine. 
\/ 3600 = 60 = tiuice the chord of half the are. 
^^ 60 X 18^ X 10 194400 ^ ^^^ ^ 
JSoiu 45^^59 ~ ^c9'9 = 4*2959, tvhich added to twice 

tJie chord of half the arc gives 64*2959 = the length of the arc. 

50 X 60 = 3000 
18 X 27 = 486 

2514 reserved number. 



AC = n/50 X 18 = 30 = the chord of half the arc. 



30 X 2 X 18 X 10 10800 ^^„^ 

2514 ~ 2 "14 ~ 4*2959, ivhich added to twice the 

chord of half the are gives 64*2959 = the length of the are. 

To find the area of a czVc?e.— Multiply half the circumference by 
half the diameter, and the product will be the area. 

Or take \ of the product of the whole circumference and diameter. 
What is the area of a circle whose diameter is 42, and circum- 
ference 131*946 ? 

2) 131*946 

65*973 = J circumference. 
21 = J diameter. 



65973 
131946 



1385*433 = area required. 
What is the area of a circle whose diameter is 10 feet 6 inches, 
and circumference 31 feet 6 inches ? 
fe. in. 

15 9 = 15*75 = J circumference. 
5 3 = 5*25 = |- diameter. 
7875 
3150 
7875 



52*6875 
12 



8*2500 

^2 feet 8 inches. 

Multiply the square of the diameter by -7854, and the product 
will be the area ; or, 

Multiply the square of the circumference by '07958, and the 
product will be the area. 

The following table will also show most of the useful problems 
relating to the circle and its equal or inscribed square. 

Diameter X *8862 = side of an equal square. 

Circumf. X -2821 = side of an equal square. 

Diameter x *7071 = side of the inscribed square. 



MENSURATION OF SUPERFICIES. 51 

Circumf. X '2251 = side of the inscribed square. 
Area X -6366 = side of the inscribed square. 
Side of a square X 1-4142 = diam. of its circums. circle. 
Side of a square X 4*443 = circumf. of its circums. circle. 
Side of a square X 1*128 = diameter of an equal circle. 
Side of a square X 3 '545 = circumf. of an equal circle. 
What is the area of a circle whose diameter is 5 ? 
7854 

25 = square of the diameter. 
39270 
15708 



19-6350 = the answer. 

To find the area of a sector^ or that part of a circle ivhich is 
hounded hy any tivo radii and their included arc. — Find the length 
of the arc, then multiply the radius, or half the diameter, by the 
length of the arc of the sector, and half the product will be the 
area. 

If the diameter or radius is not given, add the square of half the 
chord of the arc, to the square of the versed sine of half the arc ; 
this sum being divided by the versed sine, will give the diameter. 

The radius AB is 40, and the chord BO 

of the whole arc 50, required the area of d 

the sector. .-^^^^^^^^ ~"^^\ 



80 - x/802 - 50^ . ^.. . 
Q = 8* < 750 = the versed 

sine of half the arc. 



80 X 60 - 8-7750 x 27 = 4563-0750 
the reserved number. 



2 X v/8-7750 X 80 = 52-9906 = twice the 
chord of half the arc. 

52-9906x8*7750x10 ,^^^ , , ,,, . . . 
.^/.o.QhrrQ = 1*0190, which added to twice the chord 

of half the arc gives 54*0096 the length of the arc. 
54*0096 X 40 
And 2 — 1080*1920 = area of the sector required. 

As 360 is to the degrees in the arc of a sector, so is the area of 
the whole circle, whose radius is equal to that of the sector, to the 
area of the sector required. 

For a semicircle, a quadrant, &c. take one half, one quarter, &c. 
of the whole area. 

The radius of a sector of a circle is 20, and the degrees in its 
arc 22 ; what is the area of the sector ? 

Here the diameter is 40. 

Rence, the area of the circle = 40^ X *7854 = 1600 X *7854 = 
1256*64. 

^''olv, 360° : 22° ; .* 1256*64 : 76-7947 = area of the sector. 



52 



THE PRACTICAL MODEL CALCULATOR. 



To find the area of a segment of a circle. — Find the area of 
tlie sector, having the same arc Tvith the segment, by the last pro- 
blem. 

Find the area of the triangle formed by the. chord of the seg- 
ment, and the radii of the sector. 

Then the sum, or difference, of these areas, according as the 
segment is greater or less than a semicircle, will be the area re- 
quired. 

The difference between the versed sine and radius, multiplied by 
half the chord of the arc, will give the area of the triangle. 

The radius OB is 10, and the chord AC 10 ; 
what is the area of the segment ABC 1 c 

^^ AC^ 100 ^ ^ 

CD = TvcC = "oTT = 5 = the versed sine 



CE 
of half the are. 



20 



20 X 60 - 5 

Qiumher. 



X 27 = 1065 = the reserved 



10 X 2 X 5 X 10 



•9390, and this added 




1065 
to tivice the chord of half the arc gives 20-9390 = the length of the arc. 

20-9390 X 10 ^ ^^ 
^ = 104-6950 = area of the sector OACB. 

OD = 00 = CD = 5 the perpendicular height of the triangle. 
AD = n/AO^ - OD^ = ^/75 = 8-6603 = I the chord of the arc. 
8-6603 X 5 = 43-3015 = the area of the triangle AOB. 
104-6950 — 43-3015 = 61*3935 = area of the segment required; 
it being in this case less than a semicircle. 

Divide the height, or versed sine, by the diameter, and find the 
quotient in the table of versed sines. 

Multiply the number on the right hand of the versed sine by the 
square of the diameter, and the product will be the area. 

When the quotient arising from the versed sine divided by the 
diameter, has a remainder or fraction after the third place of deci- 
mals ; having taken the area answering to the first three figures, 
subtract it from the next following area, multiply the remainder by 
the said fraction, and add the product to the first area, then the 
sum will be the area for the whole quotient. 

If the chord of a circular segment be 40, its versed sine 10, and 
the diameter of the circle 50, what is the area ? 
5-0)1-0 

•2 = tabular versed sine. 
•111823 = tabular segment. 
2500 = square of 50. 
55911500 
223646 
279-557500 = area required. 



MENSURATION OF SUPERFICIES. 



53 



To find the area of a circular zone, or the space included hetiveen 
any tioo parallel chords and their intercepted arcs. — From the 
greater chord subtract half the difference between the two, mul- 
tiply the remainder by the said half difference, divide the product 
by the breadth of the zone, and add the quotient to the breadth. 
To the square of this number add the square of the less chord, and 
the square root of the sum will be the diameter of the circle. 

Now, having the diameter EG, and the two chords AB and DC, 
find the areas of the segments ABE A, and DCED, the difference 
of which will be the area of the zone required. 

The difference of the tabular segments multiplied by the square 
of the circle's diameter will give the area of the zone. 

When the larger segment AEB is greater than a semicircle, find 
the areas of the segments AGB, and DCE, and subtract their sum 
from the area of the whole circle : the remainder will be the area 
of the zone. 

The greater chord AB is 20, the less DC 15, 
and their distance Dr 17 J : required the area 
of the zone ABCD. 

20-15 

■ 2 = 2'5 = J = the difference hetiveen 

the chords. 
lT-5 + (^ 
20 = DF. 



2-5) X 2-5 

Tr5 



= 17-5 + 2-5 




And >/202 + 15^ = v/625 = 25 = the diameter of the circle. 

The segment AEB heing greater than a semicircle, we find tlie 
versed sine o/DCE = 2-5, and that o/AGB = 5. 

2-5 

Hence -^^ — 400 = tabular versed sine of DEC. 

5 
And or = '200 = tabular versed sine of AGB. 

Now -040875 X 25^ = area of seg. DEC = 25-546875 
And -111823 x 25^ = area of seg. AGB = 69-889375 

sum 95-43625 
•7854 X 25^ = area of the whole circle, = 490-87500 
Difference = area of the zone ABCD = 395*43875 

To find the area of a circular ring, or the 
space included between the circumference of 
two concentric circles. — The difference between 
the areas of the two circles will be the area of 
the ring. 

Or, multiply the sum of diameters by their 
difference, and this product again by -7854, 
and it will give the area required. 

The diameters AB and CD are 20 and 15 : required the area of 
e2 




54 



THE PRACTICAL MODEL CALCULATOR. 



tlie circular ring, or the space included between tlie circumfe- 
rences of those circles. 




Here AB -f CD X AB - CD = 35 X 5 = 1T5, anclVl^ X -7854 = 
137'4450 = area of the ring required. 

To find the areas of lunes, or the spaces between the intersecting 
arcs of two eccentric circles. — Find the areas of the two segments 
from which the lune is formed, and their difference will be the 
area required. 

The following property is one of the most curious : 

If ABC be a right angled triangle, 
and semicircles be described on the three 
sides as diameters, then will the said tri- 
angle be equal to the two lunes D and F 
taken together. 

For the semicircles described on AC and 
BC = the one described on AB, from each 
take the segments cut off by AC and BC, then will the lunes AFCE 
and BDCGr = the triangle ACB. 

The length of the chord AB is 40, the 
height DC 10, and DE 4: required the 
area of the lune ACBEA. 

The diameter of the circle oftvhich ACB 
20^ + 102 
is a part = ^tk — = 50. 

And the diameter of the circle ofivhich AEB 
= 104. 



171-8842 




Now having the diameter and versed sines., we find, 

The area of seg. ACB = -111823 x 50^ = 279-5575 
Andareaofseg.KEB = -009955 X 104^ = 107-6733 
Their difference is the area of the lune \ _ 
AEBCA required, j ~~ 

To fimd the area of an irregular polygon, or a figure of any 
number of sides. — Divide the figure into triangles and trapeziums, 
and find the area of each separately. 

Add these areas together, and the sum will be equal to the area 
of the whole polygon. 

Required the area of the irre- 
gular figure ABCDEFGA, the fol- 
lowing lines being given : 
GB = 30-5 An = 11% CO = 6 
GD = 29 Fg=ll Cs = 6-6 
FD = 24-8 E^; =4 



Here 



An + Co 



xGB = 



11-2 + 6 



2 -— 2 

X 30-5 + 8-6 X 30-5 = 262-3 = 
area of the trapezium ABCG. 




DECIMAL APPROXIMATIONS. 



55 



Also, 



And ?1±^ X GD = ll^^ X 29 = 8-8 x 29 = 255-2 = 

area of the trajyeziwn GCDF. 

FD X Ep 24-8 X 4 99-2 ^^ ^ ^ , . 
2 ~ 2 ~ ~9~~ '^^'^ ~ '^^'^'^ o/Me triangle 

FDE. 

TfAenee 262-3 + 255-2 + 49-6 = 567-1 = area of the luhole 
figure required. 

DECIMAL APPROXIMATIONS FOB, FACILITATINa CALCULATIONS IN 
MENSURATION. 



Lineal feet multiplied by 


•00019 


= 


miles 


. 


— yards 


— 


•000568 


= 


— 




Square inches 


— 


•007 


= 


squai 


•e feet. 


— yards 


— 


•0002067 


' = 


acres 


. 


Circular inches 


— 


•00546 


= 


square feet. 


Cylindrical inches 


— 


•0004546 


; = 


cubic feet. 


— feet 


— 


•02909 


= 


cubic 


yards. 


Cubic inches 


— 


•00058 


= 


cubic 


feet. 


— feet 


— 


•03704 


= 


cubic 


yards. 


— — 


— 


6-232 


= 


impel 


ial gallons. 


— inches 





•003607 


= 




— 


Cylindrical feet 


— 


4-895 


= 




— 


— inches 


— 


•002832 


= 




— 


Cubic inches 





•263 


= 


S)s. avs. of cast iron, 


— 





•281 


= 





wrought do. 


— 


— 


•283 


= 


— 


steel. 


— 


— 


•3225 


= 


— 


copper. 


— 


— 


•3037 


= 


— 


brass. 


— 


— 


•26 


= 


— 


zinc. 


— 


— 


•4103 


= 


— 


lead. 


— 


— 


•2636 


= 


— 


tin. 


— 


— 


•4908 


= 


— 


mercury. 


Cylindrical inches 


— 


•2065 


= 


— 


cast iron. 


— 


— 


•2168 


= 


— 


wrought iron. 


— 


— 


•2223 


= 


— 


steel. 


— 


— 


•2533 


= 


— 


copper. 


— 


— 


•2385 


= 


— 


brass. 








•2042 


= 





zinc. 


— 


— 


•3223 


= 


— 


lead. 


— 


— 


•207 


= 


— 


tin. 


— 


— 


•3854 


= 


— 


mercury. 


Avoirdupois lbs. 


— 


•009 


= 


cwts. 




— 


— 


•00045 


= 


tons. 




183-346 circular inches 




= 


1 square foot. 


2200 cylindrical inches 




= 


1 cabic foot. 


French metres X 3-281 




== 


feet. 




— kilogrammes 


! X 2-205 


= 


avoirdupois lb. 


— grammes X 


•002205 


= 


avoirdupois lbs. 



56 



THE PRACTICAL MODEL CALCULATOR. 



Diameter of a sphere X '806 = dimensions of equal cube. 

Diameter of a sphere X ^6667 = length of equal cylinder. 

Lineal inches X '0000158 = miles. 

A Erench cubic foot = 2093*47 cubic inches. 

Imperial gallons X '7977 = New York gallons. 

The average quantity of water that falls in rain and snow at 
Philadelphia is 36 inches. 

At West Point the variation of the magnetic needle, Nov. 16th, 
1839, was 7° 58' 27'' West, and the dip 73° 26' 28". 



DECIMAL EQUIVALENTS TO FRACTIONAL PARTS OF LINEAL 
MEASURES. 



One inch, the integer or whole number. 


•96875 i + ^ 


•625 1 


: ^^8125 1 + ^ 


•9375 1 + ^ 


•59375 1 + ^ 


•25 1 


•90625 1 + i. 


•5625 ^ + ^ 


•21875 1 + ,1 


•875 B 1 


•53125 -2 i + ^ 


•1875 S 1 + ^ 


•84375 ^ f + ^ 


•5 ^ i- 


•15625 ^1 i + A 


•8125 &^f + ^ 


•46875 fi + ^ 


•125 g. i 


•78125 : 1 + ^ 


•4375 : 1 + A 


•09375 I ^ 


•75 U 1 


•40625 ^ f + a\ 


•0625 ^ ^ 


•71875 1 + ^ 


•375 1 


' ^03125 i^ 


•6875 1 + ^ 


•34375 l + ^l 


•65625 1 + 3^, 


•3125 1 + ^ ii 


One foot, or 12 inches, the integer. 


•9166 - 11 inches.' -4166 ^ 5 inches. 'i ^0625 ^ f of in. 


•6338 ^10 — 11 -3333 ^ 4 — 


1 -05208 -^ f — 


•75 1 9 — -25 ^ 3 — 


•041661 1 — 


•QQQQ ^ S — t •1666 r2 — 


-03125 n — 


•5833 o 7 — !! -0833 p 1 — 


-02083 o 1 _ 


•5 c^ 6 — li -07291 ^ 1 — 


-01041 ^ i — 


One yard, or 36 inches, the integer. 


•9722 35 inches. 


•6389 23 inches. 1 -3055 llinches.| 


•9444 34 — 


•6111 22 — 


-2778 10 — 


•9167 33 — 


•5833 21 — 


-25 9 — 


•8889 o 32 — 


•5556 ^ 20 — 


•2222 o 8 — 


•8611 :: 31 — 


•5278:^19 — 


.1944- 7 _ 


•8333 g 30 — 


•5 1 18 — 


-1667 g 6 — 


•8056^29 — 


•4722^17 — 


•1389 r 5 — 


•7778 £ 28 — 


•4444 S 16 — 


•1111 o 4 _ 


•75 =^ 27 — 


•4167 ^ 15 — 


•0833 =^ 3 — 


•7222 26 — 


•3889 14 — 


•0555 2 — 


•6944 25 — 


•3611 13 — ■} -0278 1 — 


•6667 24 — 


•3333 12 — 



Tcible containing the Circumferences, Squares, Cubes, and Areas of 
Circles, from 1 to 100, advancing by a tenth. 



Diara 


Circum 


Square. 


Cube. 


Area. 


Diam. 


Circum. 


Square. 


Cube. 


Area. 


1 


3-1416 


1 


1 


-7854 


9 


28-2744 


81 


729 


63-6174 




•1 


3-45o7 


1-21 


1-331 


•9503 


-1 


28-5885 


82-81 


753-571 


65-0389 




•2 


3-7699 


1-44 


1-728 


1-1309 


•2 


28-90-27 


84-64 


778-688 


66-4762 




•3 


4-0840 


1-69 


2-197 


1-3273 


•3 


29-2168 


80-49 


804-357 


67-9292 




•4 


4-3982 


1-96 


2-744 


1-5393 


•4 


29-5310 


88-36 


830-584 


69-3979 




•5 


4-7124 


2-25 


3-375 


1-7671 


-5 


29-8452 


99-25 


857-375 


70-8823 




•6 


5-0265 


2-56 


4-096 


2-0106 


-6 


301593 


92-16 


884-736 


72-3824 




7 


53407 


2-89 


4-913 


2-2698 




30-4735 


94-09 


912-673 


73-8982 




•8 


5-6548 


3-24 


5-832 


2-5446 


•8 


30-7876 


96-04 


941-192 


75-4298 




9 


5-9690 


3-61 


6-859 


2-8352 


•9 


31-1018 


98-01 


970-299 


76-9770 


2 




6-2832 


4 


8 


3-1416 


10 


31-4160 


100 


1000 


78-5400 




1 


6-5973 


4-41 


9-261 


3-4636 


•1 


31-7301 


102-01 


1030-301 


80-1186 




2 


6-9115 


4-84 


10-648 


3-8013 


•2 


320443 


104-04 


1061-208 


81-7130 




3 


7-2256 


5-29 


12-167 


4-1547 


•3 


32-3580 


106-09 


1092-727 


83-3230 




4 


7-5398 


5-76 


13-8-24 


4-5239 


•4 


32-6726 


108-16 


1124-864 


84 9488 




6 


7-8540 


6-25 


15-625 


4-9087 


-6 


32-9S6S 


110-25 


1157-6-25 


86 5903 




6 


8-1681 


6-76 


17-576 


5-3093 


•6 


33-3009 


112-36 


1191-016 


88-2475 




7 


8-48-23 


7-29 


19-683 


5-7255 




33 6151 


114-49 


1225-043 


89-9204 




8 


8-7964 


7-84 


21-952 


6-1575 


•8 


33-9292 


116-64 


1259-712 


91-6090 




9 


91106 


8-41 


24-389 


6-6052 


-9 


34-2434 


118-81 


1295 029 


93-3133 


3 




9-4'248 


9 


27 


7-0686 


11 


34-5576 


121 


1331 


95-0334 




1 


9-7389 


961 


29-791 


7-5476 


•1 


34-8717 


123-21 


1367-631 


96-7691 




2 


100531 


10-24 


32-768 


80424 


•2 


35-1859 


125-44 


1404-928 


98-5205 




3 


10-3672 


10-89 


35-937 


8-5530 


•3 


35-5010 


127-69 


1442-897 


100-2S77 




4 


10-6814 


11-56 


39-304 


9-0792 


•4 


35-8142 


129-90 


1481-544 


102-0705 




5 


10-9956 


1225 


42-875 


9-6211 


•5 


36-1284 


132-25 


15-20-875 


103-8091 




6 


11 3097 


12-96 


46-656 


10-1787 


•6 


36-4425 


134-56 


1560-896 


105-6834 




7 


11-6239 


13-69 


50-653 


10-7521 


•7 


36-7567 


136-89 


1601-613 


107-5134 




8 


11-9380 


1444 


54-872 


11-3411 


•8 


37-0708 


139-24 


1643-032 


109-3590 




9 


12-2522 


15-21 


59-319 


11-9459 


-9 


37-3840 


141-61 


1685-159 


111-2204 


4 




12-5664 


16 


64 


12-5664 


12 


37-6992 


144 


1728 


113-0976 




1 


12-8805 


16-81 


68-921 


132025 


•1 


380133 


146-41 


1771-561 


114-9904 




2 


13-1947 


17-64 


74-088 


13-8544 


•2 


38-3275 


148-84 


1815-848 


116-8989 




3 


13-5088 


18-49 


79-507 


14-5220 


•3 


38-6416 


151-29 


1860-867 


118-8231 




4 


13-8-230 


19-36 


85-184 


15-2053 


•4 


38-9558 


153-76 


1906-6-24 


120-7631 




5 


14-1372 


20-25 


91-1-25 


15-9043 


-5 


39-2700 


156-25 


1953-125 


12-2-7187 




6 


14-4513 


21-16 


97-336 


16-6190 


•6 


39-5841 


158.76 


2000-376 


124-0901 




7 


14-7055 


22-09 


103-823 


17-3494 


•7 


39-8983 


161-29 


2048-383 


1-26-6771 




8 


15-0796 


2304 


110-592 


18-0956 


-8 


40-2124 


163-84 


2097-152 


1-28 6799 




9 


15-3938 


2401 


117-649 


18-8574 


-9 


40-5266 


166-41 


2146-689 


130-6984 


5 




15-7080 


25 


125 


19-6350 


13 


40-8408 


169 


2197 


132-73-26 




1 


160221 


2601 


132-651 


20-42S2 


•1 


41-1549 


171-61 


2248-091 


134-7824 




2 


16-3363 


27-04 


140-608 


21-2372 


•2 


41-4091 


174-24 


2299-968 


136-8480 




3 


16-6504 


28-09 


148-877 


2-2-0618 


•3 


41-7832 


176-89 


2352-637 


138-9294 




4 


16-9646 


29-16 


157-464 


22-9022 


•4 


42-0974 


179-56 


2406 104 


141-0264 




5 


17-2788 


30-25 


166-375 


23-7583 


-6 


42-4116 


182-25 


2460-375 


143-1391 




6 


17-5929 


31-36 


175-616 


24-6301 


•6 


42-7257 


184-96 


2515-456 


145-2675 




7 


17-9071 


32-49 


185-193 


25-5176 


•7 


43-0399 


187-69 


2571-353 


147-4117 




8 


18-2212 


33-64 


195-112 


26-4208 


•8 


43-3540 


190-44 


26-28-072 


149 5715 




9 


18-5354 


34-81 


205-379 


27-3397 


-9 


43-6682 


193-21 


2685-619 


151-7471 


6 




18-8496 


36 


216 


28-2744 


14 


43-9824 


196 


2744 


153-9384 




1 


19-1637 


37-21 


226-981 


29-2-247 


-1 


44-2965 


198-81 


2803-221 


156-1453 




2 


19-4779 


38-44 


2:38-328 


30-1907 


•2 


44-6107 


201-64 


2863-288 


158-3680 




3 


19-79-20 


39-69 


250-047 


31-1725 


-3 


44-9248 


204-49 


29-24-207 


160-0064 




4 


20-1062 


40-96 


262-144 


32-1699 


•4 


45-2390 


207-36 


2985-984 


162-8605 




5 


20-4204 


42-25 


274-6-25 


33-1831 


-5 


45-5532 


210-25 


3048-625 


165-1303 




6 


20-7345 


43-56 


287-496 


34-2120 


•6 


45 S673 


213-16 


3112-136 


167-4158 




7 


21-0487 


44-89 


300-763 


35-2566 


-7 


46-1815 


216-09 


3176-523 


169-7179 




8 


21-3628 


46-24 


314-432 


36-3168 


•8 


46-4956 


219-04 


3241-792 


172-0340 




9 


21-6770 


47-61 


328-509 


37-3928 


•9 


46-8098 


222-01 


3307-949 


174-3666 


7 




21-9912 


49 


343 


38-4846 


15 


47-1240 


225 


3375 


176-7150 




1 


22-3053 


50-41 


357-911 


39-5920 


•1 


47-4381 


228-01 


3442-951 


179-0790 




2 


22-6195 


51-84 


373-248 


40-7151 


•2 


47-7523 


231-04 


3511-808 


181-4588 




3 


229336 


53-29 


389-017 


41-8539 


-3 


48-0664 


234-09 


3581-577 


183-8542 




4 


23-2478 


54-76 


405-224 


430085 


•4 


48-3806 


237-16 


3652-264 


186-2654 




5 


23-56-20 


56-25 


421-875 


44-1787 


-5 


48-6948 


240-25 


3723 875 


188-6923 




6 


23-8761 


57-76 


438-976 


45-3647 


•6 


49-0089 


243-36 


3796-416 


191-1349 




7 


24-1903 


59-29 


456-533 


46-5663 


•7 


49-3231 


246-49 


3869-893 


193-5932 




8 


24-5044 


60-84 


474-552 


47-7837 


•8 


49-6372 


249-64 


3944-312 


190-0672 




9 


24-8186 


62-41 


493-039 


49-0168 


•9 


49-9514 


252-81 


4019-679 


198-5569 


8 




25-1328 


64 


512 


50-2656 


16 


50-2656 


256 


4096 


201-0624 




1 


25-4469 


65-61 


531-441 


51-5300 


•1 


505797 


259-21 


4173-281 


203-5835 




2 


25-7611 


67-24 


551-368 


5-2-8102 


•2 


50-8939 


262-44 


4251-528 


206-T209 




3 


26-0752 


68-89 


571-787 


64-1062 


•3 


61-2080 


265-69 


4330-747 


208-67-23 




4 


26-3894 


70-56 


692-704 


55-4178 


•4 


51-5224 


268-96 


4410-944 


211-1411 




5 


26-7036 


72-25 


614125 


66-7451 


-5 


51-8364 


272--25 


4492-125 


213-8251 




6 


27-0177 


73-96 


636-056 


68-0881 


•6 


52-1505 


275-56 


4574-296 


210-4248 




7 


27-3319 


75-69 


658-503 


59-4469 


•7 


52-4647 


278-89 


4657-463 


219 0402 




8 


27-6460 


77-44 


681-472 


60-8213 


•8 


527788 


282-24 


4741-032 


221-0712 


•9 


27-9602 


79-21 


704-90'J 


62-2115 


-9 


53-0930 


285-61 


4826-809 


2-24-3180 



57 



58 



THE PRACTICAL MODEL CALCULATOR. 



Diam- 


Circum. 


Square. 


Cube. 


Area. 


Diam. 


Circum. 


Square. 


Cube. 


Area. 


17 


53-4072 


289 


4913 


226-9806 


25 


78-5400 


625 


15625 


490-8750 


•1 


53-7213 


292-41 


5000-211 


229-6588 




1 


78-8541 


630-01 


15813-251 


494-8098 


•2 


540355 


295-84 


5088-448 


232-3527 




2 


79-1683 


635-04 


16003-008 


498-7604 


•3 


54-3496 


299-29 


6177-717 


235-0623 




3 


79-4824 


640-09 


16194-277 


502-7266 


•4 


64-6038 


302-76 


5268-024 


237-7877 




4 


79-7966 


645-16 


16387-064 


506-7086 


•0 


549780 


306-25 


5359-375 


240-5287 




5 


80-8108 


650-25 


16581-376 


510-7063 


•6 


55-2921 


309-76 


5451-776 


243-2855 




6 


80-4249 


655-36 


16777-216 


514-7196 


•7 


65-6063 


313-29 


5545-233 


2460579 




7 


80-7391 


660-49 


16974-593 


518-7488 


•8 


55-9204 


316-84 


5639-752 


248-8461 




8 


81-0532 


665-64 


17173-512 


522-7936 


•9 


56-2346 


320-41 


5735-339 


251-6500 




9 


81-3674 


670-81 


17373-979 


520-8541 


18 


56-5488 


324 


5832 


254-4696 


26 




81-6816 


676 


17576 


630-9304 


•1 


56-8629 


327-61 


6929-741 


257-3048 




1 


81-9976 


681-21 


17779-581 


535-02-23 


•2 


57-1771 


331-24 


€028-568 


260-1558 




2 


82-3099 


686-44 


17984-728 


639-1299 


•3 


57-4912 


33489 


6128-487 


2630226 




3 


82-6240 


691-69 


18191-447 


643-2533 


•■I 


57-8054 


338-56 


6229-504 


265.9050 




4 


82-9382 


696-96 


18399-744 


547-3923 


•5 


58-1196 


342-25 


6331-625 


268-8031 




5 


83-2524 


702-25 


18609-625 


551-5471 


•6 


68-4337 


345-96 


6434-856 


271-7169 




6 


83-5665 


707-56 


18821-096 


555-7176 


•7 


68-7479 


349-69 


6539-203 


274-6465 




7 


83-8807 


712-89 


19034-163 


559-9038 


•8 


59-0620 


353-44 


6644-672 


277-5917 




8 


84-1948 


718-24 


19248-832 


564-1056 


9 


59-3762 


357-21 


6751-269 


280-55-27 




9 


84-5090 


723-61 


19465-109 


568-3232 


19 


59-6904 


361 


6859 


2S.3-5294 


27 




84-8232 


729 


19683 


572-5566 


•1 


60-0045 


364-81 


6967-871 


286-5217 




1 


85-1373 


734-41 


1990-2-511 


576-8056 


•2 


60-3187 


368-64 


7077-888 


289-5298 




2 


85-4515 


739-84 


20123-648 


581-0703 


•3 


60-6328 


372.49 


7189-057 


292-5536 




3 


85-7656 


745-29 


20346-417 


585-3507 


■i 


60-9470 


376.36 


7301-384 


295-5931 




4 


86-0798 


750-76 


20570824 


589-6469 


•5 


61-2612 


380-25 


7414-875 


298-6483 




6 


86-3940 


756-25 


20796-875 


593-9587 


•6 


61-5753 


38416 


7529-536 


301-7192 




6 


86-7081 


761-76 


21024-576 


598-2863 


•7 


61-8895 


388-09 


7645-373 


3048060 




7 


87-02-23 


767-29 


21253-933 


602-6295 


•S 


62-2036 


392-04 


7762-392 


307-9082 




8 


87-3364 


772-84 


21484-952 


606-9885 


•9 


62-5178 


396-01 


7880-599 


311-0252 




9 


87-6506 


778-41 


21717-639 


611-3632 


20 - 


62-83-20 


400 


8000 


314-1600 


28 




87-9648 


784 


21952 


615-7536 


•1 


63-1461 


404-01 


8120-601 


317-3094 




1 


88-2789 


789-61 


22188-041 


620-1596 


•2 


63-4003 


40804 


8242-408 


320-4746 




2 


88-5931 


795-24 


22425-768 


624-5814 


•3 


63-7744 


412-09 


8365-427 


323-6554 




3 


88-9072 


800-89 


22665-187 


629-0190 


■-L 


64-0886 


416-16 


8489-664 


326-8520 




4 


89-2-214 


806-56 


22906-304 


633-4722 


•5 


64-40-28 


420-25 


8615-125 


3.30-0643 




5 


89-5356 


812-25 


23149-126 


637-9411 


•6 


647161 


4-24-36 


8741-816 


333-2923 




6 


89-8497 


817-96 


23393-656 


642-4257 


•7 


65-0311 


428-49 


8869-743 


336-5360 




■7 


901639 


823-69 


23639-903 


646-9261 


•8 


65-3452 


432-G4 


8998-912 


339 7954 




8 


90-4780 


829-44 


23887-872 


651-4421 


•9 


65-6594 


436-81 


9129-329 


343-0705 




9 


90-7922 


835-21 


24137-569 


655-9739 


21 


65-9736 


441 


9261 


346-3614 


29 




91-1064 


841 


24389 


660-5214 


•1 


66-2870 


445-21 


9393-931 


349-6679 




1 


91-4205 


846-81 


24642-171 


665-0845 


•2 


66-6012 


449-44 


9528-128 


352-9901 




2 


91-7347 


852-64 


24S97-0S8 


669-6634 


•3 


66-7916 


453-69 


9663-597 


356-3281 




3 


9-2-0488 


858-49 


25153-757 


674-2580 


•4 


67-2930 


457-96 


9800-344 


359-6817 




4 


92-3630 


864-36 


25412-184 


678-8683 


•5 


67-5444 


462-25 


9938-375 


363 0511 




5 


92-6772 


870-25 


25072-375 


683-4943 


•6 


67-8585 


466-56 


10077-696 


366-4362 




6 


929913 


876-16 


25934-330 


688-1360 


•7 


68-1727 


470-89 


10218-313 


369-8370 




7 


93-3055 


882-09 


26198-073 


69279.34 


•s 


68-4868 


475-24 


10360-232 


373-2534 




8 


93-6196 


888-04 


28463-592 


697-4666 


•9 


68-8010 


479-61 


10503-459 


376-6856 




9 


93-9338 


89401 


26730-899 


702-1554 


22 


6ti-1152 


484 


10648 


380-1336 


30 




94-2480 


900 


27000 


706-8600 


•1 


09-4293 


488-41 


10793-861 


383-5972 




1 


94-5621 


906-01 


27270-901 


711-5802 


•2 


69-7435 


492-84 


10941-048 


387-0765 




2 


94-8763 


912-04 


27543-608 


716-3162 


•3 


700576 


497-29 


11089-567 


390-5751 




3 


95-1904 


918-09 


27818-127 


721-0678 


•4 


70-3718 


501-76 


112.39-4-24 


394-0823 




4 


95-5046 


924-16 


28094-464 


725-8352 


•5 


70-6860 


506-25 


11390-625 


397-6087 




5 


95-8188 


930-25 


28372-625 


730-6183 


•G 


710001 


510-76 


11.543-176 


401-1509 




6 


96-1329 


936-36 


28652-616 


735-4171 


"7 


71-3143 


515-29 


11697-083 


404-7087 




7 


96-4471 


942-49 


28934-443 


740-2316 


•8 


71-6284 


519-84 


11852-352 


408-2823 




8 


96-7612 


948-64 


29218-112 


745-0618 


•9 


71-94-26 


524-41 


12008-989 


411-8716 




9 


97-0754 


954-81 


29503-629 


749 9077 


23 


7-2-2568 


529 


1-2167 


415-4766 


31 




97-3896 


961 


29791 


754-7694 


•1 


72-5709 


533-61 


12326-391 


419-0972 




1 


97-7037 


967-21 


30080-231 


759-6467 


•2 


7-2-8S51 


. 538-24 


1-2487-168 


422-7336 




2 


98-0179 


973-44 


30371-328 


764-5397 


•3 


73-1992 


542-89 


12649-337 


426-3858 




3 


98-3320 


979-69 


30664-297 


769-4485 


•4 


73-5134 


547-56 


1-2812-904 


430-0536 




4 


98-6452 


985-96 


30959-144 


774-3729 


•5 


73-8-276 


55-2-25 


12977-875 


4337371 




5 


98-9604 


992 25 


31255-875 


779-3131 


•6 


741417 


556-96 


13144-256 


437-4363 




6 


99-2745 


998-56 


31554-496 


784-2689 


.7 


74-4559 


561-69 


13312-053 


441-1511 




7 


99-5887 


1004-89 


31855-013 


789-2406 


•8 


74-7680 


566-44 


13481-272 


444-8819 




8 


99-9028 


1011-24 


32157-432 


794-2278 


•9 


75-0SS2 


571-21 


13651-919 


448-6283 




9 


100-2170 


1017-61 


32461-759 


799-2308 


24 


75-3984 


576 


138-24 


452-3904 


32 




100-5312 


10-24 


32768 


804-2496 


•1 


75-7125 


580-81 


13997-541 


456-1681 




1 


100-8453 


1030-41 


33076-161 


809-2840 


•2 


76-0267 


585-64 


1417-2-488 


459-9616 




2 


101-1595 


1036-84 


38386-248 


814-3341 


•3 


76-3408 


590-49 


14348-907 


463-7708 




3 


101-4736 


1043-29 


33698-267 


819-3999 


•4 


76-652! 


595-36 


145-26-784 


467-5957 




•4 


101-7478 


1049-76 


34012-2-24 


824-4815 


•5 


76-9692 


600-25 


14706-125 


471-4363 




5 


10-2-1020 


1056-25 


34328-125 


829-5787 


•G 


77-2833 


605-16 


14886-936 


475-2926 




6 


102-4161 


1962-76 


34645-976 


8346917 


•7 


77-5975 


610-09 


15069-2-23 


479-1646 




•7 


102-7303 


1069-29 


34965-783 


839-8203 


•8 


77-9116 


61504 


1.^252-992 


483-0524 




•8 


103-0444 


1075-84 


35287-552 


844-9647 


•9 


78-2258 


620-01 


15438-249 


486-955S 


.9 


103-3586 


1082-41 


35611-289 


850-1248 



CIRCLES, ADYAXCING BY A TENTH. 



59 



Diam. 


Circum. 


Square. 


Cube. 


Area. 


Diam. 


Circum. 


Square. 


Cube. 


Area. | 


33 


103-6728 


1089 


35937 


855-3006 


41 


128-8056 


1681 


68321 


1320-2574 




1 


103-9N69 


1095-61 


36261-691 


860-4920 


■1 


129-1197 


1689-21 


69426-531 


1326-7055 




2 


104-3011 


1102-24 


36594-368 


865-6992 


-2 


129-4323 


1697-44 


69934-528 


13331C93 




3 


104-6151 


1108-89 


36926-037 


870-9222 


•3 


129-7480 


1705-69 


70444-937 


1339-6489 




4 


104-9294 


1115-56 


37259-704 


876-1608 


•4 


130-0622 


1713-96 


70957-944 


1.346-1441 







105-2436 


1122-25 


37595-375 


881-4151 


•5 


130-3764 


1722-25 


71473-375 


1352-6551 




6 


105-5577 


1128-96 


37933-056 


886-6851 


•6 


130-6905 


1730-56 


71991-296 


1359-1818 




r 


105-8719 


1135-69 


38272-753 


891-9709 


•7 


131-0047 


1738-89 


72511-713 


1365-7242 




8 


106-1860 


1142-44 


38614-472 


897-2723 


•8 


131-3188 


1747-24 


73034-632 


1372-2822 




9 


106-5002 


1149-21 


38958-219 


902-5895 


-9 


131-6320 


1755-61 


73560-059 


1378-8500 


34 




106-8144 


1156 


39304 


907-9224 


42 


131-9472 


1764 


74088 


13S5-4456 




1 


107-1285 


lie-2-81 


39651-821 


913-2709 


•1 


132-2613 


1772-41 


74618-461 


1392-0508 




2 


107-4-272 


1169-64 


40001-688 


918-6352 


•2 


132-5755 


1780-84 


75151-448 


1398-G717 




3 


107-7568 


1176-49 


40353-C07 


924-0115 


•3 


13-2-8896 


1789-29 


75G86-967 


1405-3083 




4 


108-0710 


1183-36 


40707-584 


923-4109 


•4 


133-2038 


179776 


76225-024 


1411-9C07 




5 


108-3852 


1190-25 


41063-6-25 


934-82-23 


•5 


133-5180 


1806-25 


76765-625 


1418-6287 




6 


108-6993 


1197-16 


41421-736 


940-2494 


-6 


133-8.321 


1814-76 


77308 776 


1425-3125 




7 


109-0352 


1204-09 


41781-923 


945-6922 


•7 


134-1463 


1823-29 


77854-483 


1432-0119 




8 


109-3076 


1211-04 


42144-192 


951-1508 


-8 


134-4604 


1831-84 


78402-752 


1438-7271 




9 


109-6418 


1218-01 


42508-649 


956-6250 


-9 


134-7746 


1840-41 


78358-5S9 


1445-4580 


35 




109-9560 


1225 


4-2875 


962-1150 


43 


1350888 


1849 


73-'.07 


1452-2048 




1 


110-2701 


1232-01 


43243-551 


967-6206 


•1 


135-4029 


1857-61 


80002-991 


1458-9668 




2 


110-584O 


1239-04 


43614-208 


973-1420 


•2 


135-7171 


1866-24 


80621-568 


1465-7448 




3 


110-8984 


1246-09 


43986-977 


978-6790 


-3 


1360332 


1874-89 


81182-737 


1472-5385 




4 


111-2126 


1253-16 


44361-864 


984-2318 


•4 


136-3454 


1883-56 


81746-504 


1479-3480 




5 


111-526S 


1260-25 


44738-875 


989-8003 


-5 


136-6596 


1892-25 


8231-2->75 


1486-1731 




b 


111-8409 


1267-36 


45118-016 


995-3845 


•6 


136-9737 


1900-96 


82881-856 


1433-0139 




7 


11-2-1551 


1274-49 


45499-293 


1000-9843 


-7 


137-2879 


1909-09 


83453-4.J3 


1499-8705 




8 


112-4692 


1281-64 


45882-712 


1006-6000 


•8 


137-6020 


1918-44 


840-27-672 


1506-7427 




9 


112-7834 


12S8-S1 


4626S-279 


1012-2313 


-9 


137-9162 


19-27-21 


84604-519 


1513-0287 


36 




113-0976 


1296 


46656 


1017-8784 


44 


138-2304 


1936 


85184 


1520-5344 




1 


113-4117 


1303-21 


47045-831 


1023-5411 


•1 


138-5445 


1944-81 


85766-121 


1627-4537 




2 


113-7259 


1310-44 


47437-928 


10-29-2195 


•2 


138-8587 


1953-64 


86350-888 


1534-3883 




3 


114-0400 


1317-69 


47832-147 


1034-9131 


•3 


139-17-28 


1962-49 


86938-307 


1541-3306 




4 


114-3542 


1324-96 


48228-544 


1040-6235 


•4 


139-4870 


1971-36 


87528-384 


1548-3001 




5 


114-6684 


1332-25 


48027-125 


1046-3491 


•5 


139-8012 


1980-25 


88121-125 


1555-2883 




6 


114-9825 


1339-56 


49027-896 


1052-0904 


•6 


140-1153 


1989-16 


88716-636 


156228C2 




7 


115-2367 


1346-89 


49430-863 


1057-8474 


•7 


140-4295 


1998-09 


89314-623 


15C9-23C8 




8 


115-6108 


1354-24 


49836-032 


1063-6-200 


-8 


140-7436 


2007-04 


89915-392 


1576-3-292 




9 


115-3-250 


1361-61 


50-243-409 


1069-4084 


•9 


141-0578 


•201601 


90518-849 


1583-3742 


37 




116-2332 


1369 


50653 


1075-2126 


45 


141-3720 


2025 


91125 


1590-4350 




1 


116-5533 


1376-41 


51064-811 


1081-0324 


•1 


141-6861 


2034-01 


91733-851 


1597-5114 




2 


116-8675 


1383-84 


51478-848 


10868679 


-2 


142-0003 


2043-04 


92345-408 


] 604-6036 




3 


117-1816 


1391-29 


51895-117 


1092-7191 


•3 


142-3144 


2052-09 


92359-677 


1011-7114 




4 


117-4958 


1398-76 


52313-624 


1098-5862 


-4 


142-6286 


2061-16 


9.3576-664 


1618-8360 




5 


117-8100 


1406-25 


5-2734-375 


1104-4687 


-5 


142-9428 


2070-25 


94196-375 


1G25-9743 




6 


118-1241 


1413-76 


53157-376 


1110-3671 


-6 


143-2569 


2079-36 


94818-816 


1633-1293 




7 


118-4383 


1421-29 


53582-633 


1116-2S11 


•7 


143-5711 


2088-49 


95443-993 


1640-3020 




8 


118-75-24 


1428-84 


54010-152 


112-2-2109 


•8 


143-8852 


2097-84 


96071-912 


1647-4SG4 




9 


119-0666 


1436-41 


54439-939 


1128-1564 


•9 


144-1994 


2106-81 


96702-579 


1654-0885 


38 




110-3808 


1444 


54872 


11341176 


46 


144-5136 


2116 


97336 


1G61-90C4 




1 


119-6949 


1451-61 


55306-341 


1140-0946 


-1 


144-8277 


2125-21 


97972-181 


1669-1399 




2 


120-0091 


1459-24 


55742-968 


1146-0870 


•2 


145-1419 


2134-44 


98611-128 


1G76-S891 




3 


120-3232 


1466-89 


56181-887 


11.52-0954 


•3 


145-4560 


2143-69 


99252-847 


1683-6541 




4 


1-20-6374 


1474-56 


56623-104 


1158-1194 


•4 


145-7702 


2152-96 


99897-344 


1690-9347 




5 


1-20-9516 


1482-25 


57066-6-25 


1164-1.591 


-5 


146-0844 


2162-25 


100544-625 


1G98-2311 




(i 


121-2657 


1489-96 


5751-2-456 


1170-2145 


•6 


146-3985 


2171-56 


101194-696 


1705-5432 




7 


1-21-5799 


1497-69 


57960-603 


1176-2857 


•7 


146-71-27 


2180-89 


101847-563 


1712 8710 




8 


121-8940 


1505-44 


58411072 


1182-3725 


•8 


147-0268 


2190-24 


102503-232 


1720-2144 




9 


122-2082 


1513-21 


58863-869 


1188-4651 


•9 


147-3410 


2199-61 


103161-709 


1727-5736 


39 




122-5-224 


1521 


59319 


1294-5.394 


47 


147-6552 


2209 


1038-23 


1734-9486 




1 


122 8365 


1528-81 


59776-471 


1200-7273 


•1 


147-9693 


2218-41 


104487-111 


1742-3.392 




2 


123-1507 


1536-64 


60236-288 


1206-8770 


•2 


148-2835 


2227-84 


105154-048 


1749-7455 




3 


123-4648 


1544-49 


60698-457 


12130424 


•3 


148-5976 


2237-29 


1058-23-817 


1757-1675 




4 


123-7790 


1552-36 


61162-984 


1219-2-243 


•4 


148-9118 


2246-76 


106496-4-24 


1704-6045 




5 


124-0932 


1560-25 


61629-875 


1225-4-203 


•5 


149-2260 


2256-25 


107171-875 


17720587 




6 


124-4073 


1568-16 


62099-136 


1231-6328 


•6 


149-5361 


2265-76 


107850-176 


1779-5-279 




7 


124-7215 


1576-09 


6-2570-773 


1237-8610 


-7 


149-8543 


2275-29 


108631-.333 


1787-01-27 




•8 


125-0356 


1584-04 


63044-792 


1244-1210 


•8 


150-1684 


2-284-84 


109215-352 


1794-5133 




•9 


125-3498 


1592-01 


63521-199 


1250-3646 


-9 


150-4826 


2294-41 


109902-239 


1802-0296 


40 




125-6640 


1600 


64000 


1256-6400 


48 


150-7968 


2.304 


110592 


1809-5616 




1 


1-25-9781 


1608-01 


64481-201 


1262-9310 


•1 


151-1109 


2313-61 


111284-641 


1817-1092 




2 


126-2923 


1016-04 


64964-808 


1269-2388 


•2 


151-4251 


2323-24 


111980-168 


1824-G726 




3 


126-6064 


1624-09 


65450-827 


1275-5602 


•3 


151-7392 


2.332-89 


112678-587 


183-2-2518 




4 


120-9206 


1632-16 


65939-264 


1281-8984 


•4 


152-05.34 


2342-56 


1113379-904 


1839-84G6 




5 


127-2348 


1640-25 


664.30-1-25 


1288-2523 


•5 


152-3676 


2352-25 


114084-125 


1847-4571 




6 


1-27-54S9 


16i8-3G 


6692.3-416 


1294-6-219 


•6 


15-2-6817 


2361-96 


114791-256 


1855-0833 




7 


127-8631 


1656-49 


67419143 


1301-0071 




15'2-9959 


2371-69 


115501-303 


186-2-7253 




8 


128-1772 


1664-64 


67317-312 


1.307-4082 


•8 


153-3100 


2381-44 


110-214-272 


1S70-.38-29 


•9 


1-28-4914 


1672-81 


68417-929 


1313-8249 


-9 


153-6-242 


2391-21 


116030169 


1878-0563 



60 



THE PRACTICAL MODEL CALCULATOR. 



DiaiH. 


Circum. 


Square. 


Cube. 


Area. 


Diam. 1 Circum. 


Square. 


Cube. 


Area. 


49 


153-9384 


2401 


117649 


1885-7454 


57 


179-0712 


3249 


1S5193 


2551-7646 




1 


154-2525 


2410-81 


118370-771 


1893-4501 


•1 


179-3853 


3260-41 


186163-411 


2560-7200 




2 


154-5667 


21:-20-64 


119095-488 


1901-1706 


•2 


179-6995 


3271-84 


187149-248 


25C9-7031 




3 


154-S80S 


2430-49 


119823-157 


19o8-9u68 


•3 


180-0136 


3283-29 


188132-517 


2578-6959 




4 


155-1950 


2440-36 


120553-784 


1916-65S7 


•4 


180-3-278 


3-294-76 


189119-224 


2587-7045 




5 


155-5092 


2450-25 


121287-375 


1924-4263 


•5 


180-64-20 


3306-25 


190109-375 


2596-7287 




6 


155-8-233 


2460-16 


122023-936 


1932-2096 


•6 


180-9561 


3317-76 


191102-976 


2605-7687 




7 


15(i-lo75 


2470-09 


122763-473 


1940-0086 


•7 


181-2803 


3329-29 


192100-033 


2614-8243 




8 


156-4516 


2480-04 


123505-992 


1947-8234 


•8 


181-5844 


3-340-84 


193100-552 


2623-8957 




9 


156-7558 


2490-01 


124251-499 


1955-6538 


•9 


181-8986 


3352-41 


194104-539 


2632-9828 


50 




157-0800 


2500 


125000 


1963-5000 


58 


18-2-2128 


3364 


195112 


264-2-0856 




1 


157-3941 


2510-01 


125751-591 


1971-3618 


•1 


182-5269 


3375-61 


196122-941 


2651-2046 




2 


157-7083 


2520-04 


126506-008 


1979-2394 


•2 


182 8411 


3387-24 


197137-368 


2660-3382 




3 


158-0224 


2530-09 


127263-5-27 


19S7-1326 


•3 


183-1552 


3398-89 


198155-287 


2669-4882 




4 


158-3366 


2540-16 


128024-064 


1995-0416 


•4 


183-4694 


3410-56 


199176-704 


2678-6538 




5 


158-6508 


2550-25 


128787-625 


2U02-9663 


•5 


183-7836 


3422-25 


200201-625 


2687-8351 




6 


15S-9649 


2560-36 


1-29554-216 


2010-9067 


•6 


184-0977 


3433-96 


201230 056 


2697-0321 




7 


159-2791 


2570-49 


130323-843 


2018-S62S 


•7 


184-4119 


3445-69 


202262-003 


2706-2449 




8 


159-5932 


2580-64 


131096-512 


2026-8346 


•8 


184-7260 


3457-44 


203297-472 


2715-47-33 




9 


159-9074 


2590-81 


lol872-229 


2034-8770 


•9 


185-0402 


3469-21 


204336-469 


2724-7175 


51 




160 2216 


2601 


132651 


2042-6254 


59 


185-3544 


3481 


205379 


2733-9774 




1 


160-5357 


2611-21 


133432-831 


2050-8443 


•1 


185-66S5 


3492-81 


206425-071 


2743-2529 




2 


160-8499 


2621-44 


134217-7-28 


2058-8784 


•2 


185-9827 


3504-64 


207474-688 


275-2-5442 




3 


161-1640 


2631-69 


135005-697 


20669293 


•3 


186-2696 


3516-49 


•208527-8-57 


2761-8512 




4 


161-4782 


2641-96 


135796-744 


2074-9953 


•4 


186-6110 


3528-36 


209584-584 


2771-1739 




5 


161-7924 


2652-25 


136590-875 


2083-0771 


•5 


186-9252 


3540-25 


210644-875 


2780-51-23 




6 


162-1065 


2662-56 


137388-096 


2091-1746 


•6 


187-2393 


3552-16 


211708-736 


2789-8664 




7 


162-4207 


2672-89 


138188-413 


2099-2878 


•7 


187-5535 


3564-09 


212776-173 


2799-2362 




8 


162-7348 


2083-24 


13S991-832 


2107-4166 


•8 


187-8676 


3576-04 


213847-192 


2808-6-218 




9 


163-0490 


2693-61 


139798-359 


2115-5612 


•9 


188-1818 


3588-01 


214921-799 


2818-0-230 


52 




163-3632 


2704 


140608 


2123-7216 


60 


188-4960 


3600 


216000 


2827-4400 




1 


163-6773 


2714-41 


141420-761 


2131-8976 


•1 


188-8101 


3612-01 


217081-801 


28.36-8726 




2 


103-9935 


2724-84 


142236-648 


2140-0893 


•2 


189-1243 


3624-04 


218167-208 


2846-3210 




3 


164-3056 


2735-29 


143055-667 


2148-2967 


•3 


189-4384 


3636-09 


219256-227 


2855-7850 




4 


164-6198 


■2745-76 


143377-824 


2156-5199 


•4 


189-7526 


3648-16 


220348-864 


2865-2648 




5 


164-9340 


2756-25 


144703-125 


2164-7587 


•5 


190-0668 


3660-25 


221445-125 


2874-7G03 




6 


165-2481 


2766-76 


145531-576 


2173-0133 


•6 


190-3809 


3672-36 


222545-016 


2884-2615 




r 


165-5623 


2777-29 


146363-183 


2181-2835 


•7 


190-6951 


3684-49 


223648-543 


2893-7984 




8 


165-8764 


2787-84 


147197-952 


2189-5695 


•8 


191-0092 


3696-64 


224755-712 


2903-3410 




9 


166-1906 


2798-41 


14S035-889 


2197-8712 


•9 


191-3234 


3708-81 


225866-529 


2912-8993 


53 




166-5048 


2S09 


14S877 


2-206-1886 


61 


191-6376 


3721 


226981 


2922-47-34 




1 


166-8189 


2819-61 


149721-291 


2214-5216 


•1 


191-9517 


3733-21 


228099-131 


2932-0631 




2 


167-1331 


2830-24 


150568-768 


2222-8704 


•2 


192-2659 


3745-44 


2'29220-928 


2941-6685 




3 


107-4472 -2940-89 


151419-437 


2231-2350 


•3 


192-5800 


3757-69 


230346-397 


2951-2897 




4 


167-7614 2S51-56 


152273-304 


2239-6152 


•4 


192-8942 


3769-96 


2.31475-544 


2960-9265 




5 


16v„::-: ^-.2-25 


153130-375 


2248-0111 


•6 


193-2084 


3782-25 


232C08-375 


29705791 




6 


1H-. ^ - ;„3 


153990-656 


2256-4227 


•6 


193-5225 


3794-56 


233744-896 


29802474 




7 


U-. .: - _o-._.9 


154854153 


2264-8701 


•7 


193-8367 


3806-89 


234885-113 


2989-9314 




8 


lOj-ui.iu 


2!sy4-44 


155720-872 


2-273-2;j31 


■8 


194-1508 


38L9-24 


236029 032 


2999GSC0 




9 


169-:i322 


•2905-21 


156590-819 


2281-7519 


•9 


194-4650 


3831-61 


237176-659 


3009-3464 


54 




169-6464 


2916 


157464 


2290-2264 


62 


194-7792 


3844 


23S32S 


;:019-0776 




1 


169-9605 


2926-81 


158340-421 


2-298-7165 


•1 


195-0933 


3856-41 


239483-061 


3028-8-244 




2 


170-2747 


2937-64 


159220-0S8 


2307-2224 




195-4075 


3868-84 


240641-848 


3038-5809 




3 


170-5888 


2948-49 


160103-007 


2315-7440 


•3 


195-7216 


3881-29 


241804-367 


3048-3651 




4 


170-9030 


2959-36 


160989-184 


2324-2813 


•4 


196-0358 


3893-76 


242970-624 


3058-1591 




5 


171-2172 


2970-25 


161878-6-25 


2332-8343 


•5 


196-3500 


3906-25 


244140-6-25 


.3067-9687 




6 


171-5313 


2981-16 


162771-336 


2341-4030 


•6 


196-6641 


3918-76 


245314-376 


3077-7941 




7 


171-S455 


299-2-09 


163667-3-23 


2349-9874 


•7 


l&6-i:i7S3 


3'j31-29 


246491-883 


3087 -6341 




8 


n -21 596 


3003-04 


164566-592 


2358-5876 


•8 


!• •:-■::■■:-, •'/+3-^4 


247673-152 


3097-4919 




9 


172-4738 


3014-01 


165469-149 


2367-2034 


•9 


1,7 . ■ -6-41 


248858-189 


3107-3644 


55 




172-7880 


3025 


160375 


2375-8350 


63 


1^7 ■■ -''- ■-■• <-''J 


250047 


3117-2526 




1 


173-1U2I 


3036-01 


167284-151 


2384-4822 


•1 


198-2349 


oySl-61 


251-239-591 


3127-1564 




2 


1734163 


3047-04 


16S196-608 


2393-1452 


•2 


198-5491 


3994-24 


252435-968 


.3137-0758 




3 


173-7304 


3058-09 


169112-377 


2401-823S 


•3 


198-S632 


4006-89 


253036-137 


.3147-0114 




4 


174-0416 


3069-16 


170031-464 


2410-5182 


•4 


199-1774 


4019-56 


2-54840-104 


3156-9C64 




5 


174-3588 


3080-25 


170953-875 


2419-2283 


•5 


199-4916 


4032-25 


256047-875 


3160-9291 






174-67-29 


3091-36 


171879-616 


24-27-9541 


•6 


199-8057 


4044-96 


257259-456 


3176-9115 




7 


174-9771 


3102-49 


17-2S08 693 


2436-6956 


•7 


200-1199 


4057-69 


258474-853 


3186-9097 




8 


175-3092 


3113-64 


173741-112 


2445-4528 


•8 


•200-4340 


4070-44 


•259694-072 


3196-9235 




9 


175-0154 


3124-81 


174676-879 


2454-2257 


•9 


200-7482 


4083-21 


260917 119 


3206-9531 


56 




175-9296 


3136 


175616 


2463-0144 


64 


201-0624 


4096 


262144 


3-216 9984 




1 


176-2437 


3147-21 


176558-481 


2471-8187 


•1 


201-3765 


4108-81 


263374-721 


3227-0593 




2 


176-5579 


3158-44 


177504-328 


24S0-G3S7 


•2 


201-6907 


4121-64 


264609-288 


3237-1360 




3 


176-8720 


3169-69 


178453-547 


2489-4745 


•3 


2020048 


4134-49 


265847-707 


3-247-2-284 




•4 


177-1862 


3180-96 


179406-144 


2498-3-259 


•4 


202-3190 


4147-36 


267089-984 


3-257-3365 




5 


177-5004 


3192-25 


180362-125 


2507-1931 


•5 


202-6332 


4160-25 


268336-125 


3267-4603 




6 


177-8145 


3-203-56 


181321-496 


2516-0760 


•6 


202-9473 


4173-16 


269580-136 


3277-5998 




7 


173-1287 


3214-89 


18-2284-263 


2524-9736 


"7 


203-2615 


4186-09 


270840-0-23 


3287-75.50 




8 


178-442S 


3226-24 


183250-432 


2533-8888 


•8 


•203-575(i 


4199-04 


27-2097-792 


3297-9260 


•9 


178-7570 


3237-61 


184220-009 


2542-8188 


•9 


203-8898 


4212-01 


273359-449 


3308-1126 



CIRCLES, ADVAXCIJv^a BY A TENTH. 



61 



Diam. 


Circum. 


Square. 


Cube. 


Area. 


Diam. 


Circum. 


Square. 


Cube. 


Area. 


65 


204-2040 


4225 


274625 


3318-3150 


73 


229-,3368 


5.329 


389017 


4185-3966 




1 


2-14-5181 


4238-01 


275894-451 


3328-5340 


•1 


229-6509 


5343-61 


390617-891 


419C-S712 




2 


•2U4-8323 


4251-04 


277167-808 


3338-7668 


-2 


229-9651 


6368-24 


39-2223-168 


4-208-3014 






2J5-1464 


4264^09 


278445-077 


3349-0162 


•3 


230-2792 


5372-89 


393832-S37 


4-219-SG7S 




4 


2U5-40,J6 


4277-16 


279726-204 


3359-2814 


-4 


230-5934 


5387-66 


395446-904 


4231-3893 




5 


205-7748 


42y0-25 


281011-375 


3369-56-23 


•5 


'230-9076 


6402-25 


397066-375 


4242-9271 




6 


2j6-0.^89 


4303-36 


282300-416 


3379-8589 


•6 


231-2217 


5416-96 


398688-256 


4254-48113 






•20G-4031 


4316-49 


283593-393 


3390-1712 


•7 


231-5359 


5431-69 


400315-553 


421.6-0493 




8 


206-7172 


4329-64 


284890-312 


3400-4992 


•8 


231-8500 


6446-44 


401947-272 


427 7 -0339 




9 


•207-0ol4 


4342-81 


286191-179 


3410-8429 


•9 


232-1642 


6461-21 


403583-419 


42892343 


66 




207-3456 


4356 


287496 


3421-2024 


74 


232-4784 


6476 


405224 


4300-8504 




1 


207-6597 


4369-21 


288804-781 


3431-5776 


•1 


232-7925 


5490-81 


406869-021 


4:,12-4821 




2 


20 r -9739 


4382-44 


290117-628 


3441-9633 


•2 


233-1067 


6505-64 


40&618-4S8 


4324-1-206 






20S-2SS0 


4395-69 


291434-247 


3462-3749 


•3 


233-4208 


5620-49 


410172-407 


4335-7928 




4 


20S-6022 


4408-96 


292754-944 


3462-7971 


•4 


233-7360 


5535-36 


411830-784 


4347-4717 




6 


208-9164 


4422-25 


294079-626 


3473-2351 


-5 


234-0492 


5550-25 


413493-625 


4359-16G3 




6 


209-2305 


4435-56 


295408-296 


3483-6888 


•6 


234-3633 


6565-16 


415160-936 


4.370-8766 




y 


2U9-5447 


4448-89 


296740-963 


3494-1640 


•7 


234-6775 


5680-09 


41683-2-723 


4.382-6026 




8 


200-L>o88 


4462-24 


298077-632 


3504-6432 


-8 


234-9916 


5696-04 


418508-992 


4394-3448 




9 


210-1730 


4475-61 


299418-309 


3515-1430 


■9 


235-3058 


5610-01 


420189-749 


4406-1018 


67 




210-4S72 


4489 


30U763 


3525-6606 


75 


235-620O 


5625 


421875 


4417-8750 




1 


210-8013 


4502-41 


30-2111-711 


3536-1928 


•1 


235-9341 


5640-01 


423664-751 


4429-6G38 




2 


211-1165 


4515-84 


303i64-448 


3546-7407 


-2 


236-2483 


5655-04 


425259-008 


4441-4684 




3 


211-4296 


4529-29 


304821-217 


3557-3043 


•3 


236-5624 


567009 


420957-777 


4453-2886 




4 


211-7438 


4542-76 


300182-024 


3567-8837 


-4 


236-8766 


5685-16 


428601-064 


4465-1-246 




5 


212-0580 


4556-25 


307546-876 


3.578-4787 


•6 


237-1908 


5700-25 


4303C8-875 


4476-97G3 




6 


2123721 


4569-76 


308915-776 


3589-0895 


-6 


237-6049 


5715-36 


432081-210 


44S8-8437 




7 


2120863 


4583-29 


310288-733 


3599-7159 


•7 


237-8191 


6730-49 


43379S-093 


4o00-72G8 




8 


213-0004 


4696-84 


311666-752 


3610-3581 


-8 


238-1332 


6745-64 


435519-512 


4512-625G 




9 


213-3146 


4610-41 


313046-839 


3621-0160 


-9 


238-4474 


5760-81 


437-245-479 


4524-5401 


68 




213-6288 


4624 


314432 


3631-6896 


76 


•238-7616 


6776 


438976 


4536-4704 




1 


213 94-29 


4637-61 


315821-241 


3642-3788 


•1 


239-0757 


6791-21 


440711-081 


4548-41G3 




2 


214-2571 


4661-24 


317214-568 


3653 0838 


•2 


239-3899 


5806-44 


4424.50 728 


45o0-3787 




o 


214-5712 


4664-89 


318611-987 


3663-8040 


•3 


-239-7040 


5821-69 


444194-947 


4o7 2-35.53 




4 


214-8854 


4678-56 


3-20013-504 


3674-5410 


•4 


240-0182 


5836-96 


44.5943-744 


4584-3583 




5 


215-1996 


4692-25 


321419-125 


36S6-2931 


-6 


240-3324 


5862-25 


447697-1^26 


4596-3571 




6 


215-5137 


4705-96 


322828-856 


3696-0060 


•6 


240-6465 


5867-56 


449456-096 


4G0S-3S16 




- 


215-*279 


4719-69 


324242-703 


3706-8445 


•7 


240-9607 


6882-89 


461217-663 


4G-20-421S 




8 


21d-1420 


4733-44 


325660-672 


3717-6437 


•8 


241-2748 


5898-24 


462984-832 


4G3-2-477G 




9 


216 4562 


4747 21 


327082-769 


3728-4687 


•9 


241-5987 


5913-61 


454756-609 


4G44 5492 


69 




216-7701 


4761 


328509 


3739-2894 


77 


•241-9032 


5929 


456633 


4G56-C366 




1 


217-0845 


4774-81 


329939-371 


3750-1357 


-1 


•242^2173 


5944-41 


458314-011 


4068-7396 






217-39s7 


4788-64 


331373-888 


3700-9978 


•2 


242^5316 


5969-84 


460099-648 


4180-8583 




3 


217-7r2S 


4802-49 


332812-557 


3771-8756 


•3 


24^2-8456 


5975-29 


461889-917 


4692-9927 




4 


21,^-0270 


4816-36 


334256-384 


3782-7691 


4 


243-1598 


5990-76 


463684-824 


4705-1429 




6 


218-:J412 


4830-25 


335702-375 


3793-6783 


-5 


243-4740 


6006-25 


465484-375 


4717-3087 




6 


218-13553 


4844-16 


337153-536 


3804-6032 




243-7881 


6021-76 


467^2S8-576 


4729-4903 






218-'j(3U5 


4858-09 


338608-873 


3815-6438 


•7 


•244^1023 


6037-29 


469097-433 


4741-6875 




8 


2ly-2S36 


4872-04 


340068-392 


3826-5002 


•8 


244^4164 


6052-84 


470910-952 


4753-9r,05 




9 


21J-5078 


4886-01 


341532-099 


3837-4722 


-9 


244^7306 


6068-41 


472729-139 


4706-1292 


70 




219-9120 


4900 


343000 


3S4S-4600 


78 


245^0448 


G084 


474552 


4778-3736 




1 


2-20-22(il 


4914-01 


344472-101 


3S59-4952 


•1 


245^3589 


6099-61 


476379-541 


4790-6336 






220-5403 


4928-04 


346948-408 


S870-4S26 




245-6731 


6115--24 


478-211-768 


4802-9094 




3 


22U-S544 


4942-09 


347428-927 


3881-5174 


-3 


245^9872 


6130-89 


480048-687 


4815-2010 




4 


221-lti8G 


4956-16 


348913-664 


3892-5680 


•4 


246-3014 


6146-56 


481890-304 


4827-5082 




5 


221-48-28 


4970-25 


350402-626 


3903-6343 


•5 


246-6156 


6162-25 


483736-625 


4839-8311 







221-7969 


4984-36 


351895-816 


3914-7163 


■6 


246-9297 


6177-96 


485587-656 


4852-1697 




7 


2-2-2-1111 


4998-49 


353393-243 


3925-8140 


•7 


247-2439 


6193-69 


487443-403 


4864-5241 




8 


•222-4252 


501-2-64 


354894-912 


3936-9274 


•8 


247-5480 


6209-44 


489303-872 


4876-8973 




9 


222-73.J4 


5026-81 


356400-829 


3948-0565 


-9 


247-8722 


6225-21 


491169-069 


4889-2799 


71 




223-1 1536 


5041 


357911 


3959-2014 


79 


248-1864 


6241 


493039 


4901-6814 




1 


223-3677 


5055-21 


359425-431 


3970-3619 


-1 


248-5006 


6256-81 


494913-671 


4914-0985 




2 


22M;819 


5069-44 


300944-1-28 


3981-6381 


•2 


248-8147 


6272-64 


490793-088 


4926-5314 




3 


223-99i;0 


5083-69 


362467-097 


3992-7301 


-3 


249-1288 


0288-49 


498677-257 


4938-98-20 




4 


224-3102 


5097-96 


363994-344 


4003-9373 


•4 


249-4430 


6304-36 


500566-184 


4951-4443 




5 


224-0244 


5112-25 


3S56-26-S76 


4015-1611 


•5 


249-7572 


0320-25 


502459-875 


4963-9243 




6 


22i-J3S5 


51-26-56 


367061-696 


4026-4002 


•6 


260-0713 


6336-16 


504358-336 


4976-4840 




7 


225-2527 


5140-89 


368601-813 


4037-6550 


.7 


250-3855 


6362-09 


506201-573 


4988-9314 




8 


2-25-5668 


5156-24 


370146-232 


4048-9254 


•8 


2.50-6996 


6368-04 


608169-592 


5001-4686 




9 


2'25-8810 


5169-61 


371694-959 


4060-2116 


-9 


251-0138 


638401 


510082-399 


5014-0014 


72 




226-1952 


5184 


373-248 


4071-5136 


80 


251-3-280 


6400 


512000 


502(;-6G0O 




1 


226-5093 


5198-41 


374805-361 


4082-8332 


•1 


251-64-21 


6416-01 


513922-401 


5039-1342 




2 


226-8235 


521-2-84 


376367-048 


4094-1645 


■2 


251-9563 


0432-04 


515849-608 


6051-7242 




3 


2-27-1376 


52'27-29 


377933-067 


4105'5125 


-3 


252-2704 


6448-09 


517781-6-27 


5064-32^18 




4 


2-27-4518 


5'241-76 


379503-424 


4116-8793 


-4 


252-5846 


6464-16 


519718-464 


6076-9552 




5 


•2-27-7660 


5256-25 


381078-125 


4V28-2587 


-5 


262-8988 


6480-25 


521660-125 


5089-5883 




6 


228-0801 


5'270-76 


382657-176 


41396524 


-6 


•2,W-21'29 


6496-36 


523606-016 


5102-2411 




T 


228-3943 


5285-29 


384-240-583 


4151-0667 


-7 


253-5271 


6512-49 


5-25557-943 


5114-9096 




8 


2-28 7084 


6-299-84 


385828-352 


4162-4943 


•8 


253-8412 


6528-64 


5'27614-112 


5127-6938 


•9 


2-29-0-220 


5314-41 


387420-489 


4173-9376 


-9 


254-1554 


6544-81 


52947 5-1-29 


5140-2937 



62 



THE PRACTICAL MODEL CALCULATOR. 



Diam. 


Circum. 


Square. 


Cube. 


Area. 1 


Uiam. 1 Circum. 


Square. Cube. 


Area. 


81 


254-4696 


65G1 


531441 


5153-0094 i 


89 


•279-6024 


7921 


704969 


62-21-1.534 


•1 


254-7S37 


6577^21 


533411-731 


5165-7407 




1 


279-9165 


7938-81 


707347-971 


6-235-1413 


•2 


255-0979 


6593-44 


535387-328 


5178-4877 




2 


2S0-2307 


7956-64 


709732-288 


6'249^1450 


•3 


255-41-20 


6609-69 


537367-797 


6191-2505 




3 


280-5448 


7974-49 


712121-957 


6263-1644 


•4 


255-7262 


6625-96 


539353-144 


5204-0285 




4 


280-8590 


7992-36 


714516-984 


6277-1995 


•5 


256-0404 


6042-25 


541343-375 


5216-8231 




5 


281-1732 


8010-25 


716917-375 


6291-2035 


•6 


25G-3545 


6658-56 


543338-496 


5229-6330 




6 


281-4873 


8028-16 


719323-136 


6305-3168 


•7 


256-6687 


6674-89 


545338-513 


5242-4586 




7 


281-8825 


8046-09 


721734-273 


6319-3990 


•8 


256-9828 


6691-24 


547343-432 


5255-2998 




8 


282-1156 


8064-04 


724150-792 


6333-4970 


•9 


257-2970 


6707-61 


549353-259 


5268-1568 




9 


282-4298 


8082-01 


726572-699 


6347-6813 


82 


257-6112 


6724 


551368 


5281-0296 


90 




282-7440 


8100 


729000 


6361-7400 


•1 


2o7-9253 


6740-41 


553387-661 


5293-9180 




1 


283-0581 


8118-01 


731432-701 


6375-8850 


•2 


258-2395 


6756-84 


555412-248 


5306-8221 




2 


283-3723 


8136-04 


733870-808 


6390-0458 


•3 


258-5536 


6773-29 


557441-767 


5319-7439 




8 


28.3-6864 


8154-09 


736314-327 


6404-22-22 


•4 


258-8646 


6789-76 


559476-224 


5332-6775 




4 


284-0006 


8172^16 


738763-264 


6418-4144 


•6 


2591820 


6806-25 


561515-625 


5345-6287 




5 


584-3148 


8190^25 


741217-625 


6432-6223 


•6 


259-4961 


6822-76 


563559-976 


5358-5957 




6 


284-6289 


8208^36 


743677-416 


6446-8474 


•7 


259-8103 


6839-29 


565609-283 


5371-5983 




7 


284-9431 


8226-49 


746142-643 


6461-0852 


•8 


260-1244 


6855-84 


567663-552 


5384-5762 




8 


285-2572 


8244-64 


748613-312 


6475-3402 


•9 


260-4386 


6S72^41 


569722-789 


5397-5908 


•9 


285-5714 


8262-81 


751089-429 


6489-6109 


83 


260-7523 


6889 


571787 


5410-6206 


91 


235-SS56 


8281 


753571 


6503-8974 


•1 


261-0669 


6905-61 


573856-191 


5423 6660 




1 


286-1997 


8299-21 


756058-031 


6518-1995 


•2 


261-3811 


692-2-24 


575930-368 


5436-7272 




2 


286-5139 


8317-44 


758550-528 


6532-5173 


•3 


261-6952 


6938-89 


578009-537 


5449-8042 




3 


286-8290 


8335-69 


761048-497 


6540-8909 


•4 


262 0094 


6955-56 


580093-704 


5462-8968 




4 


287-1422 


8353-96 


7635.51-944 


6561-2081 


•5 


262-3-236 


6972-25 


582182-875 


5476-0051 




5 


287-4564 


8372-25 


766060-875 


6575-5651 


•6 


282-6376 


6988-96 


584277-056 


5489-1291 




6 


287-7705 


8390-56 


768575-296 


6589-9458 


•7 


202-9519 


7005-69 


586376-253 


5502-2689 




7 


288-0847 


8408-89 


771095-213 


6604-3222 


•8 


263-2840 


7022-44 


588480-472 


5515-4243 




8 


288-3988 


8427-24 


773620-632 


6618-7542 


•9 


263-5S02 


7039-21 


590589-719 


5528-5958 




9 


288-7130 


8445-61 


776151-559 


6633-1820 


84 


283-8944 


7056 


592704 


5541-7824 


92 




2890272 


8464 


7786S8 


0647-6256 


•1 


264-2085 


7072-81 


594823-321 


5554-9849 




1 


2S9-.3413 


8482-41 


781229-961 


6662-0848 


•2 


264-5-227 


7089-64 


596947-688 


5568-2032 




2 


289-6555 


8500-84 


783777^44S 


6676-5597 


•3 


264-8368 


7106-49 


599077-107 


5581-4372 




3 


289-9696 


8519-29 


786330^467 


6691-0161 


•4 


265-1510 


7123-36 


601-211-584 


5594-6869 




4 


290-2838 


8537-76 


788889-024 


6705-5567 


•5 


265-4652 


7140-25 


603351-125 


5607-9523 




5 


290-5980 


8556-25 


791453-125 


6720-0787 


•6 


265-7793 


7157-16 


605495-736 


56-21-2334 




6 


290-9121 


8574-76 


794022-776 


6734-6165 


•7 


266-0935 


7174-09 


607645-423 


5634-5682 




7 


291-2263 


8593-29 


796597-983 


6749-1699 


•8 


266-4078 


7191-04 


609800-192 


5647-842S 




8 


291-5404 


8611-84 


799178-752 


6763-7391 


■9 


266-7218 


7208-01 


611960-049 


5661-1710 




9 


291-8546 


8630-41 


801765-089 


6778-3-240 


85 


2'37-0360 


7225 


614125 


5674-5150 


93 




292-1688 


8649 


804357 


6792-9246 


•1 


267-3501 


7242-01 


616295-051 


5687-8746 




1 


292-4829 


8C67-61 


806954-491 


6807-5408 


•2 


267-6643 


7259-04 


618470-208 


5701-2500 




2 


292-7971 


8686-24 


809557-568 


6822-17S0 


•3 


267-9784 


7276-09 


6-20650-477 


5714-6410 




3 


293-1112 


8704-89 


812166-2.37 


6836-8206 


•4 


268-2926 


7293-16 


622835-864 


5728-0478 




4 


293-4254 


872.3-56 


814780-504 


6851-4840 


•5 


268-6068 


7310-25 


625026-375 


5741-4703 




5 


293-7396 


8742-25 


817400-375 


6866-1631 


•6 


268-9209 


7327-36 


627222016 


5754-9085 




6 


294-0537 


8760-96 


820025-856 


6880-8579 


.': 


269-2351 


7344-49 


629422-793 


5768-3624 




7 


294-3679 


8779-69 


822656-953 


6895-5685 


•8 


269-5492 


7361-64 


631628-712 


5781-8320 




8 


294-6820 


8798-44 


825293-672 


6910-2947 


•9 


269-8034 


7378-81 


633839-779 


5795-3173 




9 


294-9962 


8817-21 


827936-019 


69-25-0367 


86 


270-1776 


7396 


636056 


5808-8184 


94 




295-3104 


8836 


830584 


6939-7944 


•1 


270-4917 


7413-21 


638277-381 


5822-3351 




1 


-295-6245 


8854-81 


833237-621 


6954-5677 


•2 


270-8059 


7430-44 


640503-928 


5835-8675 




2 


295-9387 


8873-64 


835896-888 


6969-3568 


•3 


271-1200 


7447-69 


642735-647 


5849-4157 




3 


296-2436 


8892-49 


838561-807 


0984-1614 


•4 


271-4342 


7464-96 


644972-544 


5862-9795 




4 


296-5670 


8911-36 


841232-384 


6998-9821 


•5 


271-7484 


7482-25 


647214-625 


5876-5591 




5 


296-8812 


8930-25 


843908-625 


7013-8183 


•6 


272-0665 


7499-56 


649461-896 


5890-1541 




6 


297-1953 


8949-16 


846590-536 


7028-6702 


■7 


272-3767 


7516-89 


651714-363 


5903-7654 




7 


297-5095 


8968-09 


849278-123 


7043-5025 


•8 


272-6908 


7534-24 


653972-032 


5917-3920 




8 


297-8236 


8987-04 


851971-392 


7058-4180 


•9 


273-0050 


7551-61 


656234-909 


5931-0344 




9 


298-1378 


9006-01 


854670-349 


7073-3-202 


87 


273-3192 


7569 


658503 


5944-6926 


95 




298-4520 


9025 


857375 


7088-2350 


■1 


273-6333 


7586-41 


660776-311 


5958-3644 




1 


298-7661 


9044-01 


860085-351 


7103-1654 


•2 


273-9875 


7603-84 


663054-848 


597-2-0559 




2 


299-0723 


9063-04 


862801-408 


7118-1116 


•3 


•274-2616 


7621-29 


665338-617 


5985-7691 




-3 


299-3944 


9082-09 


865523-177 


7133-0734 


•4 


274-5758 


7638-76 


667627-624 


5999-4821 




4 


209-7086 


9101-16 


868250-664 


7148-0510 


•5 


274-8900 


7656-25 


669921-875 


6013-2187 




5 


300-0228 


91-20-25 


870983-875 


7163-0443 


•6 


275-2041 


7673-76 


672221-376 


6026-9711 




-6 


300-3369 


9139-36 


873722-816 


7178-0533 


•7 


275-5183 


7691-29 


674526-133 


6040-7391 




-7 


300-6511 


9158-49 


876467-493 


7193-07S0 


•8 


275-8324 


7708-84 


676836-152 


6054-5149 




8 


300-9652 


9177-64 


879217-912 ' 7-208-1184 


•9 


276-1466 


7726-41 


679151-439 


6068-32-24 




-9 


301-2794 


9196-81 


881974079 ! 72-2.3-1745 


88 


276-4008 


7744 


6S1472 


6082-1376 


96 




301-5936 


9216 


884736 


7238-2464 


•1 


276-7749 


7761-61 


683797-841 


6095-9684 




•1 


301-9077 


9235-21 


887503-681 


7253-3339 


•2 


277-08DI 


7779-24 


686128-968 


6109-8150 






302-2219 


9254-44 


890-277-128 


7268-4371 


•3 


277-4032 


7796-89 


688465-387 


6123-6774 




-3 


30-2-5360 


9273-69 


893056-347 


72S3-5561 


•4 


277-7174 


7814-56 


690807-104 


6137-5554 




•4 


302-8502 


9292-96 


895841-344 


7298-C907 


•5 


•278-0316 


7832-25 


093154-122 


6151-4491 




-5 


303-1644 


9312-25 


898632-125 


7313-8411 


•6 


•278-3457 


7849-96 


695506-456 


6165-3585 




-6 


30.3-4785 


9331-56 


901428-696 


7329-0072 


•7 


27S-6599 


7867-69 


697864-103 


6179-2837 




.■J- 


3037027 


935089 


904231-063 


7344-1890 


•8 


278-9750 


7885-44 


7002-27-072 | 61 93-2-245 




•8 


304-1068 i 9370-24 


907039-232 


7359-3864 


1 ■' 


279-2882 


7903-21 


702595-369 


1 6207-1811 




•9 


304-4210 


; 9389-61 


909853-209 


7374-5996 



TABLE OF THE LENGTH OF CIRCULAR ARCS. 



63 



Diam. 


Circum. 


Square. 


Cube. 


Area. 


Diam. 


Circum. 


Square. 


Cube. 


.». 


97 


304-7352 


9409 


912673 


7389-8286 


•6 


309-7617 


9721-96 


958585-256 


7035-6273 


•1 


3050493 


9428-41 


915498-611 


7405-0732 


•7 


310-0759 


9741-69 


961504-803 


7651-1933 


•2 


305-3ti35 


9447-84 


918330-048 


7420-3335 




310-3960 


9761-44 


964430-272 


7666-6349 


•3 


305-6776 


9467-29 


921167-317 


7435-6095 


•9 


310-7042 


9781-21 


967361-669 


7682-1623 


•4 


305-9918 


9486-76 


924010-4-24 


7450-9013 


99 


311-0184 


9801 


970-299 


7697-7054 




306-3060 


9506-25 


926859-375 


7466-2087 


•1 


311-3325 


98-20-81 


973-242-271 


7713-2&41 


•6 


306-6'201 


95-25-76 


929714-176 


7481-5319 


•2 


311-6467 


9840-64 


976191-488 


7728-8386 




306-9363 


9545-29 


932574-833 


7496-8707 


•3 


311-9608 


9860-49 


979146-657 


7744-4288 


•8 


307-2484 


9564-84 


935441-352 


7512-2253 


•4 


312-2750 


9880-36 


982107-784 


7760-0.347 


■9 


307-5626 


9584-41 


938313-739 


7527-5956 


•5 


812-5892 


9900-25 


985074-875 


7775-6563 


98 


307-8768 


9604 


941192 


7542-9816 


•6 


312-9033 


9920-16 


988047-936 


7791-2936 


•1 


308-1909 


9623-61 


944076-141 


7558-3832 


•7 


313-2175 


9940-09 


991026-973 


7806-9466 


•2 


308-5051 


9643-24 


946966-168 


7573-8006 


■8 


313-5116 


9960-04 


994011-992 


7822-6154 


•3 


308-8192 


9662-89 


949862-087 


7589-2338 


•9 


313-8458 


9980-01 


997002-999 


7838-2998 


•4 


309-1334 


9682-56 


952763-904 


7604-68-26 


100 


314-1600 


10000 


1000000 


7854-0000 


•5 


309-4476 


9702-25 


955671-625 


7620-1471 













A Table of the Length of Circular Arcs^ radius being unity. 



Degree. 


Length- 


Degree. 


Length. '• 


Min. 


Length. 


Sec. 


Length. 


1 


0-01745S3 


60 


1-0471976 


1 


0-0002909 


1 


0-000048 


2 


0-0349066 


70 


1-2217305 


2 


0-0005818 


2 


0-000097 


3 


0-0523599 


80 


1-3962634 


3 


0-0008727 


3 


0-0000145 


4 


0-0698132 


90 


1-5707963 


4 


0-0011636 


4 


0-0000194 


5 


0-0872665 


100 


1-7453293 


5 


0-0014544 


5 


0-0000242 


6 


0-1047198 


120 


2-0943951 : 


6 


0-0017453 


6 


0-0000291 


7 


0-1221730 


150 


2-6179939 


/ 


0-0020362 


7 


0-0000339 


8 


0-1396263 


180 


3-1415927 


8 


0-0023271 


8 


0-0000388 


9 


0-1570796 


210 


3-6651914 


9 


0-0026180 


9 


0-0000436 


10 


0-1745329 


240 


4-1887902 


10 


0-0029089 


10 


0-0000485 


20 


0-3490659 


270 


4-7123890 


20 


0-0058178 


20 


0-0000970 


30 


0-5235988 


300 


5-2359878 : 


30 


0-0087266 


30 


0-0001454 


40 


0-6981317 


330 


5-7595865 j 


40 


0-0116355 


40 


0-0001939 


50 


0-8726646 


360 


6-2831853 


50 


0145444 


50 


0-0002424 



Required the length of a circular arc of 37° 42' 58" ? 
30° = 0-5235988 
7° = 0-1221730 
40' = 0-0116355 
2' = 0-0020368 
50" = 0-0002424 
8'' = 0-0000388 
The length 0-6582703 required in terms of the 
radius. 

1207° Fahrenheit = 1° of Wedgewood's pyrometer. Iron melts 
at about 166° Wedgewood ; 200362° Fahrenheit. 

Sound passes in air at a velocity of 1142 feet a second, and in 
water at a velocity of 4700 feet. 

Freezing water gives out 140° of heat, and may be cooled as 
low as 20°. All solids absorb heat when becoming a fluid, and the 
quantity of heat that renders a substance fluid is termed its caloric 
of fluidity, or latent heat. Fluids in vacuo boil with 124° less 
heat, than when under the pressure of the atmosphere. 



64 



THE PRACTICAL MODEL CALCULATOR. 



Areas of the Segments and Zones of a Circle ofiuJdch tlie Diameter 
is Unity, and supposed to he divided into 1000 equal parts. 



•001 
•002 
•003 
•004 
•005 

■006 
•007 
•008 
•009 
•010 

•on 

•012 
•013 
•014 
•015 

•016 
•017 
•018 
•019 
•020 

•021 
•022 
•023 
•024 
•025 

•026 
•027 
•028 
•029 
•030 

•031 
■032 
•033 
•034 
•035 

■036 
•037 
■038 
•039 
•040 

•041 
■042 
•043 
■044 
■045 

•046 
•047 
•048 
•049 
■050 



000042 
000119 
000219 
000337 
000470 

000618 
000779 
000951 
001135 
001329 

001533 
001746 
001968 
002199 
002438 

002685 
002940 
003202 
003471 
003748 

004031 
004322 
004618 
00492] 
005230 

005546 
005867 
006194 
006527 
006865 

007209' 
007558 
007913 
008273 
008638 

009008 
009383 
009763 
010148 
010537 

010931 
011330 
011734 
012142 
012554 

012971 
013392 
013818 
014247 
014681 



001000 
002000 
003000 
004000 
005000 

006000 
007000 
008000 
009000 
010000 

011000 
011999 
012999 
013998 
014998 

015997 
016997 
017996 
018996 
019995 

020994 
021993 
022992 
023991 
024990 

025989 
026987 
027986 
028984 
029982 

030980 
031978 
032976 
033974 
034972 

035969 
036967 
037965 
038962 
039958 

040954 
041951 
042947 
043944 
044940 

045935 
046931 
047927 
048922 
049917 



Height. 



051 
052 
053 
054 
055 

056 
057 
058 
059 
060 

061 
062 
063 
064 
065 



067 



070 

071 
072 
073 
074 
075 

076 
077 
078 
079 
080 

081 
082 
083 
084 
085 

086 
087 
088 
089 
090 

091 
092 
093 
094 
095 

096 
097 
098 
099 
100 



Area of 
Segment. 



015119 
015561 
016007 
016457 
016911 

017369 
017831 
018296 
018766 
019239 

019716 
020196 
020680 
021168 
021659 

022154 
022652 
023154 
023659 
024168 

024680 
025195 
025714 
026236 
026761 

027289 
027821 
028356 
028894 
029435 

029979 
030526 
031076 
031629 
032186 

032745 
033307 
033872 
034441 
035011 

035585 
036162 
036741 
037323 
037909 

038496 
039087 
039680 
040276 
040875 



050912 
051906 
052901 
053895 
054890 

055883 
056877 
057870 
058863 
059856 

060849 
061841 
062833 
063825 
064817 

065807 
066799 
067790 
068782 
069771 

070761 
071751 
072740 
073729 
074718 

075707 
076695 
077683 
078670 
079658 

080645 
081631 
082618 
083604 
084589 

085574 
086559 
087544 
088528 
089512 

090496 
091479 
092461 
093444 
094426 

095407 
096388 
097369 
098350 
099330 



Height. 



101 
102 
103 
104 
105 

106 
107 
108 
109 
110 

111 
112 
113 
114 
115 

116 
117 
118 
119 
120 

121 
122 
123 
124 
125 

126 
127 
128 
129 
130 

131 
132 
133 
134 
135 

136 
137 
138 
139 
140 

141 
142 
143 
144 
145 

146 
147 
148 
149 
150 



041476 
042080 
052687 
043296 
043908 

044522 
045139 
045759 
046381 
047005 

047632 
048262 
048894 
049528 
050165 

050804 
051446 
052090 
052736 
053385 

054036 
054689 
055345 
056003 
056668 

057326 
057991 
058658 
059327 
059999 

060672 
061348 
062026 
062707 
063389 

064074 
064760 
065449 
066140 
066833 

067528 
068225 
068924 
069625 
070328 

071033 
071741 
072450 
073161 
073874 



AREAS OF THE SEGMENTS AND ZONES OF A CIRCLE. 



65 



Height. 


Area of Seg. 


Area of Zone. 


Height 


Area of Seg. 


Area of Zone. 


Height. 


AreaofSeg. 


Area of Zone. 


•151 


•074589 


•148674 


•206 


•116650 


•200915 


•261 


•163140 


•248608 




152 


•075306 


•149625 


•207 


•117460 


•200924 


•262 


•164019 


•249461 




153 


•076026 


•150578 


•208 


•118271 


•201835 


•263 


•164899 


•250212 




154 


•076747 


•151530 


•209 


•119083 


•202744 


•264 


•165780 


•251162 




•155 


•077469 


•152481 


•210 


•119897 


•203652 


•265 


•166663 


•252011 




156 


•078194 


•153431 


•211 


•120712 


•204559 


•266 


•167546 


•252851 




•157 


•078921 


•154381 


•212 


•121529 


•205465 


•267 


•168430 


•253704 




158 


•079649 


•155330 


•213- 


•122347 


•206370 


•268 


•169315 


•254549 




159 


•080380 


•156278 


•214 


•123167 


•207274 


•269 


•170202 


•255392 




160 


•081112 


•157226 


•215 


•123988 


•208178 


•270 


•171080 


•256235 




161 


•081846 


•158173 


•216 


•124810 


•209080 


•271 


•171978 


•257075 




162 


•082582 


•159119 


•217 


'125634 


•209981 


•272 


•172867 


•257915 




163 


•083320 


•160065 


•218 


•126459 


•210882 


•273 


•173758 


•258754 




164 


•084059 


•161010 


•219 


•127285 


•211782 


•274 


•174649 


•259591 




165 


•084801 


•161954 


•220 


•128113 


•212680 


•275 


•175542 


•260427 




166 


•085544 


•162898 


•221 


•128942 


•213577 


•276 


•176435 


•261261 




167 


•086289 


•163841 


•222 


•129773 


•214474 


•277 


•177330 


•262094 




168 


•087036 


•165784 


•223 


•130605 


•215369 


•278 


•178225 


•262926 




169 


•087785 


•165725 


•224 


•131438 


•216264 


•279 


•179122 


•263757 




170 


•088535 


•166666 


•225 


•132272 


•217157 


•280 


•180019 


•264586 




171 


•089287 


•167606 


•226 


•133108 


•218050 


•281 


•180918 


•265414 




172 


•090041 


•168549 


•227 


•133945 


•218941 


•282 


•181817 


•266240 




173 


•090797 


•160484 


•228 


•134784 


•219832 


•283 


•182718 


•267065 




174 


•091554 


•170422 


•229 


•135624 


•220721 


•284 


•183619 


•267889 




175 


•092313 


•171359 


•230 


•136465 


•221610 


•285 


•184521 


•268711 




176 


•093074 


•172295 


•231 


•137307 


•222497 


•286 


•185425 


•269532 




177 


•093836 


•173231 


•232 


•138150 


•223354 


•287 


•186329 


•270352 




178 


•094601 


•174166 


•233 


•138995 


•224269 


•288 


•187234 


•271170 




179 


•095366 


•175100 


•234 


•139841 


•225153 


•289 


•188140 


•271987 




180 


•096134 


•176033 


•235 


•140688 


•226036 


•290 


•189047 


•272802 




181 


•096903 


•176966 


•236 


•141537 


•226919 


•291 


•189955 


•273616 




182 


•097674 


•177897 


•237 


•142387 


•227800 


•292 


•190864 


•274428 




183 


•098447 


•178828 


•238 


•143238 


•228680 


•293 


•191775 


•275239 




184 


•099221 


•179759 


•239 


•144091 


•229559 


•294 


•192684 


•276049 




185 


•099997 


•180688 


•240 


•144944 


•230439 


•295 


•193596 


•276857 




186 


•100774 


•181617 


•241 


•145799 


•231313 


•296 


•194509 


•277664 




187 


•101553 


•182545 


•242 


•146655 


•232189 


•297 


•195422 


•278469 




188 


•102334 


•183472 


•243 


•147512 


•233063 


•298 


•196337 


•279273 




189 


•103116 


•184398 


•244 


•148371 


•233937 


•299 


•197252 


•280075 




190 


•103900 


•185323 


•245 


•149230 


•234809 


•300 


•198168 


•280876 




191 


•104685 


•186248 


•246 


•150091 


•235680 


•301 


•199085 


•281675 




192 


•105472 


•187172 


•257 


•150953 


•236550 


•302 


•200003 


•282473 




193 


•106261 


•188094 


•248 


•151816 


•237419 


•303 


•200922 


•283269 




194 


•107051 


•189016 


•249 


•152680 


•238287 


•304 


•201841 


•284063 




195 


•107842 


•189938 


•250 


•153546 


•239153 


•305 


•202761 


•284857 




196 


•108636 


•190858 


•251 


•154412 


•240019 


•306 


•203683 


•285648 




197 


•109430 


•191777 


•252 


•155280 


•240883 


•307 


•204605 


•286438 




198 


•110226 


•192696 


•253 


•156149 


•241746 


•308 


•205527 


•287227 




199 


•111024 


•193614 


•254 


•157019 


•242608 


•309 


•206451 


•288014 




200 


•111823 


•194531 


•255 


•157890 


•243469 


•310 


•207376 


•288799 




201 


•112624 


•195447 


•256 


•158762 


•244328 


•311 


•208301 


•289583 




202 


•113426 


•196362 


•257 


•159636 


•245187 


•312 


•209227 


•290365 




203 


•114230 


•197277 


•258 


•160510 


•246044 


•313 


•210154 


•291146 




204 


•115035 


•198190 


•259 


•161386 


•246900 


•314 


•211082 


•291925 


•205 
1 


•115842 


•199103 


•260 


•162263 


•247755 


•315 


•212011 


•292702 

1 



f2 



ee 



THE PRACTICAL MODEL CALCULATOK. 



Height. 


Area of Seg. 


Area of Zone. 


Height. 


Area of Seg. 


Area of Zone. 


Height. 


Area of Seg. 


Area of Zone. 


•316 


•212940 


•293478 


•371 


•265144 


•333372 


•426 


•318970 


•366463 


•317 


•213871 


•294252 


•372 


•266111 


•334041 


•427 


•319959 


•366985 


•318 


•214802 


•295025 


•373 


•267078 


•334708 


•428 


•320948 


•367504 


•319 


•215733 


•295796 


•374 


•268045 


•335373 


•429 


•321938 


•368019 


•320 


•216666 


•296565 


•375 


•269013 


•336036 


•430 


•322928 


•368531 


•321 


•217599 


•297333 


•376 


•269982 


•336696 


•431 


•323918 


•369040 


•322 


•218533 


•298098 


•377 


•270951 


•337354 


•432 


•324909 


•369545 


•323 


•219468 


•298863 


•378 


•271920 


•338010 


•433 


•325900 


•370047 


•324 


•220404 


•299625 


•379 


•272890 


•338683 


•434 


•326892 


•370545 


•325 


•221340 


•300386 


•380 


•273861 


•339314 


•435 


•327882 


•371040 


•326 


•222277 


•301145 


•381 


•274832 


•339963 


•436 


•328874 


•371531 


-327 


•223215 


•301902 


•382 


•275803 


•340609 


•437 


•329866 


•372019 


•328 


-224154 


•302658 


•383 


•276775 


•341253 


•438 


•330858 


•372503 


-329 


•225093 


•303412 


•384 


•277748 


•341895 


•439 


•331850 


•372983 


•330 


•226033 


•304164 


•385 


•278721 


•342534 


•440 


•332843 


•373460 


•331 


•226974 


•304914 


•385 


•279694 


•343171 


•441 


•333836 


•373933 


•332 


•227915 


•305663 


•387 


•280668 


•343805 


•442 


•334829 


•374403 


•333 


•228858 


•306410 


•388 


•281642 


•344437 


•443 


•335822 


•374868 


•334 


•229801 


•307155 


•389 


•282617 


•345067 


•444 


•336816 


•375330 


•335 


•230745 


•307898 


•390 


•283592 


•345694 


•445 


•337810 


•375788 


•336 


•231689 


-308640 


•391 


•284568 


•346318 


•446 


•338804 


•376242 


•337 


•232034 


•309379 


•392 


•285544 


•346940 


•447 


•339798 


•376692 


•338 


•233580 


•310117 


•393 


•286521 


•347560 


•448 


•340793 


•377138 


•339 


•234526 


•310853 


•394 


•287498 


•348177 


•449 


•341787 


•377580 


•340 


•235473 


^311588 


•395 


•288476 


•348791 


•450 


•342782 


•378018 


•341 


•236421 


•312319 


-396 


•289453 


•349403 


•451 


•343777 


•378452 


•342 


•237369 


•313050 


■397 


•290432 


•350012 


•452 


•344772 


•378881 


•343 


•238318 


•313778 


•398 


•291411 


•350619 


•453 


•345768 


•379307 


•344 


•239268 


•314505 


•399 


•292390 


•351223 


•454 


•346764 


•379728 


•345 


•240218 


•315230 


•400 


•293369 


•351824 


•455 


•347759 


•380145 


•346 


•241169 


•315952 


•401 


•294349 


•352423 


•456 


•348755 


•380557 


•347 


•242121 


•316673 


•402 


•295330 


•353019 


•457 


•349752 


•380965 


•348 


•243074 


•317393 


•403 


•296311 


•353612 


•458 


•350748 


•381369 


•349 


•244026 


•318110 


•404 


•297292 


•354202 


•459 


•351745 


•381768 


•350 


•244980 


•318825 


•405 


•298273 


•354790 


•460 


•352742 


•382162 


•351 


•245934 


•319538 


•406 


•299255 


•355376 


•461 


•353739 


•382551 


•352 


•246889 


•320249 


•407 


•300238 


•355958 


•462 


•354736 


•382936 


•353 


•247845 


•320958 


•408 


•301220 


•356537 


•483 


•355732 


•383316 


•354 


•248801 


•321666 


•409 


•302203 


•357114 


•464 


•356730 


•383691 


•355 


•249757 


•322371 


•410 


•303187 


•357688 


•465 


•357727 


•384061 


•856 


•250715 


•323075 


•411 


•304171 


•358258 


•466 


•358725 


•384426 


•357 


•251673 


•323775 


•412 


•305155 


•358827 


•467 


•359723 


•384786 


•358 


•252631 


•324474 


•413 


•306140 


•359392 


•468 


•360721 


•385144 


•359 


•253590 


•325171 


•414 


•307125 


•359954 


•469 


•361719 


•385490 


•360 


•254550 


•325866 


•415 


•308110 


•360513 


•470 


•362717 


•385834 


•361 


•255510 


•326559 


•416 


•309095 


•361070 


•471 


•363715 


•386172 


•362 


•256471 


•327250 


•417 


•310081 


•361623 


•472 


•364713 


•386505 


•363 


•257433 


•327939 


•418 


•311068 


•362173 


•473 


•365712 


•380832 


•364 


•258395 


•328625 


•419 


•312054 


•362720 


•474 


•366710 


•387153 


•365 


•259357 


•329310 


•420 


•313041 


•363264 


•475 


•367709 


•387469 


•366 


•260320 


•329992 


•421 


•314029 


•363805 


•476 


•368708 


•387778 


•367 


•261284 


•330673 


•422 


•315016 


•364343 


•477 


•369707 


•388081 


•368 


•262248 


•331351 


•423 


•316004 


•364878 


•478 


•370706 


•388377 


•369 


•263213 


•332027 


•424 


•316992 


•365410 


•479 


•371704 


•388669 


•370 


•264178 


•332700 


•425 


•317981 


•365939 


•480 


•372704 


•388951 



RULES rOR FINDING THE AREA OF A CIRCULAR ZONE, ETC. 67 



Height. 


Area of Seg. 


Area of Zone. 


•481 


•373703 


•389228 


•482 


•374702 


•389497 


•483 


•375702 


•389759 


•484 


•376702 


•390014 


•435 


•377701 


•390261 


•486 


•378701 


•390500 


•487 


•379700 


•390730 


•488 


•380700 


•390953 


•489 


•381699 


•391166 


•490 


•382699 


•391370 



Height. 


Area of Seg. 


Area of Zone. 


Height. 


Area of Seg. 


AreaofZoue. 


•491 


•383699 


•391564 


•496 


•388699 


•392362 


•492 


•384699 


•391748 


•497 


•389699 


•392480 


•493 


•385699 


•391920 


•498 


•390699 


•392580 


•494 


•386699 


•392081 


•499 


•391699 


•392657 


•495 


•387699 


•392229 


•500 


•392699 


•392699 



To find the area of a segment of a circle. 

Rule. — Divide the height, or versed sine, 

by the diameter of the circle, and find the 

quotient in the column of heights. 

Then take out the corresponding area, in the column of areas, 

and multiply it by the square of the diameter ; this will give the 

area of the segment. 

Required the area of a segment of a circle, whose height is 3J 
feet, and the diameter of the circle 50 feet. 

31 = 3-25; and 3-25 ^ 50 = -065. 
•065, by the Table, = -021659 ; and -021659 x 50^ = 54-147500, 
the area required. 

To find the area of a circular zone. 
Rule 1. — When the zone is less than a semi-circle, divide the 
height by the longest chord, and seek the quotient in the column 
of heights. Take out the corresponding area, in the next column 
on the right hand, and multiply it by the square of the longest chord. 
Required the area of a zone whose longest chord is 50, and height 15. 
15 - 50 = -300 ; and -300, by the Table, = -280876. 
Hence -280876 X 50^ = 702-19, the area of the zone. 
Rule 2. — When the zone is greater than a semi-circle, take the 
height on each side of the diameter of the circle. 

Required the area of a zone, the diameter of the circle being 50, 
and the height of the zone on each side of the line which passes 
through the diameter of the circle 20 and 15 respectively. 

20 -V- 50 = -400 ; -400, by the Table, = -351824 ; and -351824 X 
502 _ 879-56. 

15 ^ 50 = -300 ; -300, by the Table, =-280876 ; and -280876 x 
50^ = 702-19. Hence 879-56 + 702-19 = 1581-75. 

Approximating rule to find the area of a segment of a circle. 

Rule. — Multiply the chord of the segment by the versed sine, 
divide the product by 3, and multiply the remainder by 2. 

Cube the height, or versed sine, find how often twice the length 
of the chord is contained in it, and add the quotient to the former 
product ; this will give the area of the segment very nearly. 

Required the area of the segment of a circle, the chord being 12, 
and the versed sine 2. 

12 X 2 = 24 ; Y = 8 ; and 8 X 2 = 16. 

2^ -^ 24 = -3333. 
Hence 16 + •3333=16-3333, the area of the segment very nearly. 



68 



PEOPOPtTIONS OF THE LENGTHS OF CIRCULAE ARCS, 



Height 


Length 


Height 


Length 


Height 


Length 


Height 


Length 


Hei.h. 


Length 


of 


of 


of 


of 


of 


of 


o: 


of 


of 


Arc. 


Arc. 


Arc 


Arc. 


Arc. 


Arc. 


Arc. 


Arc. 


Arc. 


Arc. 


•100 


1-02645 


-181 


1-08519 


-261 


1-17-275 


-341 


1^28583 


-421 


1-42041 


•101 


1-02698 


•182 


1-08611 


•262 


1-17401 


-342 


r28739 


•422 


1-42222 


•102 


1-02752 


-183 


1-08704 


•263 


1-17527 


•343 


1-28895 


-423 


1-42402 


•103 


1-02806 


•184 


1-08797 


-264 


1-17655 


•344 


1-29052 


•4-24 


1-42583 


•104 


1-02860 


•185 


1-0SS90 


•265 


1-17784 


•345 


1-29209 


•425 


1-42764 


•105 


1-02914 


•186 


1-08984 


•266 


1-17912 


•346 


1-29366 


•426 


1-42945 


•106 


1-02970 


•187 


1-09079 


-267 


1-18040 


•347 


1-29523 


•427 


1-43127 


•107 


1-03026 


-188 


1-09174 


-268 


1-18162 


•348 


1-29681 


•428 


1-43309 


•108 


1-03082 


•189 


1-09269 


-269 


1-18294 


•349 


1-29839 


•429 


1-43491 


•109 


1^03139 


•190 


1-09365 


•270 


1-18428 


•350 


1-29997 


•430 


1-43673 


•110 


1-03196 


•191 


1-09461 


•271 


1-18557 


•351 


1-30156 


•431 


1-43856 


•111 


1-03254 


-192 


1-09557 


•272 


1-18688 


•352 


1-30315 


•432 


1-44039 


•112 


1-03312 


'193 


1-09654 


•273 


1-18819 


-353 


1-30474 


•433 


1-44222 


•113 


1-03371 


•194 


1-09752 


•274 


1-18969 


•354 


1-30634 


•434 


1-44405 


•lU 


1-03430 


•195 


1-09850 


-275 


1-19082 


•355 


1-30794 


•435 


1-44589 


•115 


1-03490 


•196 


1-09949 


-276 


1-19214 


•356 


1-30954 


•436 


1-44773 


•116 


1-03551 


•197 


1-10048 


-277 


1-19345 


•357 


1-31115 


•437 


1-44957 


•117 


1-03611 


•198 


1-10147 


•278 


1-19477 


•358 


1-31276 


-438 


1-45142 


•118 


1-03672 


•199 


1-10247 


-279 


1-19610 


•359 


1-31437 


•439 


1-45327 


•119 


1-03734 


•200 


1-10348 


•280 


1-19743 


•360 


1-31599 


•440 


1-45512 


•120 


1-03797 


•201 


1-10447 


•281 


1-19887 


•361 


1-31761 


•441 


1-45697 


•121 


1-03860 


•202 


1-10548 


•282 


1-20011 


•362 


1-31923 


•442 


1-45883 


•122 


1-03923 


•203 


1-10650 


•283 


1-20146 


•363 


1-32086 


•443 


1-46069 


•123 


1-03987 


•204 


1-10752 


-284 


1-20282 


•364 


1-32-249 


•444 


1-40255 


•124 


1-04051 


•205 


1-10855 


•285 


1-20419 


•365 


1-32413 


•445 


1-46441 


•125 


1-04116 


•206 


1-10958 


-286 


1-20558 


•366 


1-32577 


•446 


1-46628 


•126 


1-04181 


-207 


1-11062 


•287 


1-20696 


•367 


1-32741 


•447 


1-46815 


•127 


1-04247 


•208 


1-11165 


•288 


1-20828 


•368 


1-32905 


•448 


1-47002 


•128 


1-04313 


•209 


1-11269 


•289 


1-20967 


•369 


1-33069 


•449 


1-47189 


•129 


1-04380 


•210 


1-11374 


-290 


1-21202 


•370 


1-33234 


•450 


1-47377 


•130 


1-04447 


•211 


1-11479 


•291 


1-21239 


-371 


1-33399 


•451 


1-47565 


•131 


1-04515 


•212 


1-115S4 


•292 


1-21381 


■372 


1-33564 


•452 


1-47753 


•132 


1-04584 


-213 


1-11692 


-293 


1-21520 


•373 


1-S3730 


•453 


1-47942 


•133 


1-04652 


•214 


1-11796 


•294 


1-21658 


•374 


1-33896 


•454 


1-48131 


•134 


1-04722 


•215 


1-11904 


•295 


1-21794 


•375 


1-34063 


•455 


1-48320 


•135 


1-04792 


•216 


1-12011 


•296 


1-21926 


•376 


1-34229 


•456 


1-48509 


•136 


1-04862 


•217 


1-12118 


•297 


1-22061 


•377 


1-34396 


•457 


1-48699 


•137 


1-04932 


•218 


1-12225 


•298 


1-22203 


•378 


1-34563 


•458 


1-48889 


•138 


1-05003 


-219 


1-12334 


•299 


1-22347 


•379 


1-34731 


•459 


1-49079 


•139 


1-05075 


•220 


1-12445 


•300 


1-22495 


•380 


1-34899 


•460 


1-49269 


•140 


1-05147 


•221 


1-12556 


•301 


1-22635 


•SSI 


1-35068 


•461 


1-49460 


•141 


1-05220 


•222 


1-12663 


•302 


1-22776 


•382 


1-35237 


•462 


1-49651 


•142 


1-05293 


•223 


1-12774 


•303 


1-22918 


•383 


1-35406 


•463 


1-49842 


•14S 


1-05367 


•224 


1-12885 


•304 


1-23061 


•384 


1-35575 


•464 


1-50033 


•144 


1-05441 


•225 


1-12997 


-305 


1-23205 


•385 


1-35744 


•465 


1-50-224 


•145 


1-05516 


•226 


1-13108 


-306 


1-23349 


•386 


1-35914 


•466 


1-50416 


•146 


1-05591 


•227 


113-219 


-307 


1-23494 


•387 


1-36084 


•467 


1-50608 


•147 


1-056C7 


•228 


1-13331 


•308 


1-23636 


•388 


1-36254 


•468 


1-50800 


•148 


1-05743 


-229 


1-13444 


•309 


1-2.3780 


•389 


1-36425 


•469 


1^0992 


•149 


1-05819 


•230 


1-13557 


•310 


1-23925 


•390 


1-36596 


•470 


1-51185 


•150 


1-05896 


•231 


1-13671 


•311 


1-24070 


•391 


1-36767 


•471 


1-51378 


•151 


1-05973 


•232 


1-13786 


-312 


1-24216 


•392 


1-36939 


•472 


1-51571 


•152 


1-06051 


•233 


1-13903 


-313 


1-24360 


•393 


1-37111 


•473 


1-51764 


•153 


1-06130 


•234 


1-14020 


•314 


1-Z4.506 


•394 


1-37283 


•474 


1-51958 


•154 


1-06209 


•235 


1-14136 


-315 


1-24654 


•395 


1-37455 


•475 


1-52152 


•155 


1-06288 


•236 


1-14247 


-316 


1-24801 


•396 


1-37628 


•476 


1-52346 


•156 


1-06368 


•237 


1-14363 


•317 


1-24946 


•397 


1-37801 


•477 


1-52541 


•157 


1-06449 


•238 


1^144S0 


•318 


1-25095 


•398 


1-37974 


•478 


1-52736 


•158 


1-065.30 


•239 


1-14597 


-319 


1-25-243 


•399 


1-38148 


•479 


1-6-2931 


•159 


1-06611 


•240 


1-14714 


•3-20 


1-25.391 


•400 


1-38322 


•480 


1-63126 


•160 


1-06693 


•241 


1-14S31 


•321 


1-25539 


•401 


1-38496 


•481 


1-53322 


•161 


1-06775 


•242 


1-14949 


•322 


1-25686 


•402 


1-38671 


•482 


1-53518 


•162 


1-06S58 


•243 


1-15007 


•3-23 


1-25836 


•403 


1-38846 


-483 


1-53714 


•163 


1-06941 


-244 


1-15186 


-324 


1-25987 


-404 


1-39021 


•484 


1-63910 


•164 


1-07025 


•245 


1-15308 


-325 


1-261.37 


■405 


1-39196 


■485 


1-64106 


•165 


1-07109 


•246 


1-154-29 


-326 


1-26286 


•406 


1-39372 


•486 


1-54302 


•166 


1-07194 


•247 


1-15549 


-327 


1-26437 


•407 


1^39548 


•487 


1-54499 


•167 


1-07279 


•248 


1-15670 


-328 


1-2658.S 


•408 


1-39724 


•488 


1-64696 


•168 


1-07365 


-249 


1-15791 


-329 


1-26740 


•409 


1-39900 


-489 


1-54893 


•169 


107451 


•250 


1-15912 


•330 


1-26892 


•410 


1-40077 


•490 


1-55090 


•170 


1-07537 


•251 


116033 


•331 


1-27044 


-411 


1-402&4 


•491 


l-55'288 


•171 


1-076-24 


•252 


1-16157 


-332 


1-27196 


•412 


1-40432 


•492 


1-55486 


•172 


1-07711 


•253 


1-16279 


-3-33 


1-27349 


•413 


1-40610 


•493 


1-55685 


•173 


1-07799 


•254 


1-16402 


•334 


1-27502 


•414 


■ 1-40788 


•494 


1-55854 


•174 


1-07888 


•255 


1-16526 


•335 


1-27656 


•415 


1-40966 


•495 


1^56083 


•175 


1-07977 


•256 


1-16649 


•336 


1-27810 


-416 


1-41145 


-496 


1-56282 


•176 


1-08066 


•257 


1-16774 


•337 


1-27864 


•417 


1-41324 


•497 


1-56481 


•177 


1-08156 


•258 


1-16899 


•338 


1-28118 


•418 


1-41503 


•498 


1-56680 


•178 


1-08246 


•259 


1-17024 


-339 


1-28273 


•419 


1-41682 


•499 


1-56879 


•179 


1-08337 


•260 


1-17150 


•340 


1-28428 


•420 


1-41861 


•500 


1-57079 


•180 


1-0S428 



















PROPORTIONS OF THE LENGTHS OP SEMI-ELLIPTIC ARCS. 



69 



PROPORTIONS OF THE LENGTHS OF SEMI- 
ELLIPTIC ARCS. 



Height 


Length of 


Height 


Length of 


Height 


Length of 


Height 


Length of 


Height 


Length of 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Are. 


of Arc. 


Arc. 


•100 


1-04162 


•157 


1^10113 


-214 


1-66678 


•271 


1-23835 


-328 


1.31472 


•101 


1-04262 


•158 


1-10224 


•215 


1-16799 


•272 


1-23966 


-329 


1-31610 


•102 


1-04362 


•159 


1-10335 


•216 


1-16920 


•273 


1-24097 


-330 


1-81748 


•103 


1-04462 


•160 


1-10447 


•217 


1-17041 


•274 


1-24228 


-331 


1-31886 


•104 


1-04562 


•161 


1-10560 


•218 


1-17163 


•275 


1-24359 


-882 


1-32024 


•105 


1.04662 


•162 


1-10672 


•219 


1-17285 


•276 


1-24480 


-338 


1-82162 


•106 


1-04762 


•163 


1-10784 


•220 


1-17407 


•277 


1-24612 


•334 


1-32300 


•107 


1-04862 


•164 


1-10896 


•221 


1-17529 


•278 


1-24744 


•385 


1-82488 


•108 


1-04962 


•165 


1-11008 


•222 


1-17651 


•279 


1-24876 


•836 


1-32576 


•109 


1-05063 


•166 


1-11120 


•223 


1-17774 


•280 


1-25010 


•387 


1-32715 


•110 


1-05164 


•167 


1-11232 


•224 


M7897 


•281 


1-25142 


•388 


1-82854 


•111 


1-05265 


•168 


1-11344 


•225 


M8020 


•282 


1-25274 


'339 


1-32993 


•112 


1-05366 


•169 


1-11456 


•226 


M8143 


•283 


1-25406 


•340 


1-83132 


•113 


1-05467 


•170 


1^11569 


•227 


1^18266 


•284 


1-25538 


•341 


1-38272 


•114 


1-05568 


•171 


M1682 


•228 


1-18390 


•285 


1-25670 


•342 


1-83412 


•115 


1-05669 


•172 


1-11795 


•229 


1-18514 


•286 


1-25803 


•348 


1-33552 


•116 


1-05770 


•173 


1-11908 


•230 


1-18638 


•287 


1-25936 


•344 


1-38692 


•117 


1-05872 


•174 


1-12021 


•231 


1-18762 


•288 


1-26069 


•345 


1 •33838 


•118 


1-05974 


•175 


1-12134 


•232 


1-18886 


■289 


1-26202 


•846 


1-83974 


•119 


1-06076 


•176 


1-12247 


•233 


1-19010' 


•290 


1-26335 


•847 


1-34115 


•120 


1-06178 


•177 


1-12360 


•234 


1-19184: 


•291 


1-26468 


•848 


1-34256 


•121 


1-06280 


•178 


1-12473 


•235 


1-19258 


•292 


1-26601 


•349 


1-34397 


•122 


1-06382 


•179 


1-12586 


•236 


1-19382 


•293 


1-26784 


•850 


1-34589 


•123 


1-06484 


•180 


1-12699 


•237 


1^19506 


•294 


1-26867 


-851 


1-34681 


•124 


1-06586 


•181 


1-12813 


•238 


1-19630' 


•295 


1-27000 


•352 


1-34823 


•125 


1-06689 


•182 


1-12927 


•239 


1-19755' 


•296 


1-27133 


•853 


1-34965 


•126 


1-06792 


•183 


1-13041 


•240 


1-19880; 


•297 


1-27267 


•354 


1-35108 


•127 


1-06895 


•184 


1-13155 


•241 


1-20005: 


•298 


1-27401 


•355 


1-85251 


•128 


1-06998 


•185 


1-13269 


•242 


1-20130' 


•299 


1-27535 


•356 


1-35394 


•129 


1-07001 


•186 


1-13383 


•243 


1-20255: 


•300 


1-27669 


•357 


1-35537 


•130 


1-07204 


•187 


1-13497 


•244 


1^20380 : 


•301 


1-27808 


•358 


1-35680 


•131 


1-07308 


•188 


1-13611 


•245 


1^20506 ; 


•302 


1-27937 


.359 


1-35823 


•132 


1-07412 


•189 


1-13726 


•246 


1-20632 


•303 


1-28071 


•360 


1-35967 


•133 


1-07516 


•190 


1-13841 


•247 


1-20758: 


•304 


1-28205 


•361 


1-36111 


•134 


1-07621 


•191 


1-13956 


•248 


1-20884: 


•305 


1-28339 


•862 


1-36255 


•135 


1-07726 


•192 


1-14071 


•249 


1-21010: 


•306 


1-28474 


•863 


1-36399 


•136 


1-07831 


•193 


1-14186 


•250 


1-21136 1 


•307 


1-28609 


•364 


1-36543 


•137 


1-07937 


•194 


1-14301 


•251 


1-21263 


•308 


1-28744 


-365 


1-36688 


•138 


1-08043 


•195 


1-14416 


•252 


1-21390 


•309 


1-28879 


-366 


1-36883 


•139 


1-08149 


•196 


1-14531 


•253 


1-21517 


•310 


1-29014 


-867 


1-36978 


•140 


1-08255 


•197 


1-14646 


•254 


1.21644 


•311 


1-29149 


•368 


1-37128 


•141 


1-08362 


•198 


1-14762 


•255 


1-21772 


-312 


1-29285 


•869 


1-87268 


•142 


1-08469 


•199 


1-14888 


•256 


1-21900 


-313 


1-29421 


-870 


1-87414 


•143 


1-08576 


•200 


1-15014 


•257 


1-22028 


•314 


2-29557 


•371 


1-37662 


•144 


1-08684 


•201 


1-15131 


•258 


1-22156 


•315 


1-29608 


•372 


1-87708 


•145 


1-08792 


•202 


1-15248 


•259 


1-22284 


•316 


1-29829 


•373 


1-87854 


•146 


1-08901 


•203 


1-15366 


•260 


r22412! 


•317 


1-29965 


•374 


1-38000 


•147 


1-09010 


•204 


1-15484 


•261 


1^22541 1 


•318 


1-30102 


•375 


1-38146 


•148 


1-09119 


•205 


1-15602 


•262 


1^22670: 


•319 


1-30289 


•376 


1-38292 


•149 


1-09228 


-206 


1-15720 


•263 


1^22799' 


•320 


1-80876 


•377 


1-88439 


•150 


1-09330 


-207 


1-15838 


•264 


1^22928, 


•321 


1-80513 


•378 


1-38585 


•151 


1-09448 


-208 


1-15957 


•265 


1-230571 


•322 


1-80650 


-379 


1-38732 


•152 


1-09558 


-209 


1-16076 


•266 


1-23186 


•323 


1-30787 


-380 


1-38879 


•153 


1-09669 


•210 


1-16196 


•267 


1-23315, 


•324 


1-30924 


-381 


1-39024 


•154 


1-09780 


•211 


1-16316 


•268 


1-23445 


•325 


1-31061 


-382 


1-39169 


•155 


1-09891 


•212 


1-16436 


•269 


1-23575 


•326 


1-31198 


-388 


1-39314 


456 


1-10002 


•213 


1-16557 


•270 


1-23705 


-327 


1-31335 


-884 


1-39459 



70 



THE PRACTICAL MODEL CALCULATOR. 



Height 


Length of 


Height 


Length of 


Height 


Length of 


Height 


Length of 


Height 


Length of 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


•385 


1-39605 


•447 


1-48850 


•509 


1 •58474 


-571 


1^68195 


•633 


1-78172 


•386 


1-39751 


•448 


1-49003 


•510 


1^58629 


•572 


1-683-54 


-634 


1-78335 


•387 


1-39897 


•449 


1-49157 


•511 


1-58784 


-573 


1-68513 


•635 


1-78498 


•388 


1-40043 


•450 


1-49311 


•512 


1-58940 


•574 


1-68672 


-636 


1-78660 


•389 


1-40189 


•451 


1-49465 


•513 


1-59096 


-575 


1-68831 


-637 


1^78823 


•390 


1-40335 


•452 


1-49618 


•514 


1-59252 


-576 


1-68990 


-638 


1^78986 


•391 


1-40481 


•453 


1-49771 


•515 


1^59408 


-577 


1-69149 


•639 


1-79149 


'392 


1-40627 


•454 


1-49924 


•516 


1 •59564 


-578 


1-69308 


•640 


1-79312 


•393 


1-40773 


•455 


1-50077 


•517 


1^59720 


•579 


1-69467 


•641 


1-79475 


•394 


1-40919 


•456 


1-50230 


•518 


1^59876 


•580 


1-69626 


-642 


1-79638 


•395 


1 •41065 


•457 


1-50383 


•519 


1^60032 


•581 


1-69785 


•643 


1-79801 


•396 


1^41211 


•458 


1-50536 


•520 


1^60188 


-582 


1-69945 


•644 


1-79964 


•397 


1-41357 


•459 


1-50689 


•521 


1^60344 


-583 


1-70105 


•645 


1-80127 


•398 


1-41504 


•460 


1-50842 


•522 


1-60500 


•584 


1-70264 


•646 


1-80290 


•399 


1-41651 


•461 


1-50996 


•523 


1-60656 


•585 


1-70424 


•647 


1-804-54 


•400 


1-41798 


•462 


1-51150 


•524 


r60812 


•586 


1-70584 


•648 


1-80617 


•401 


1-41945 


•463 


1-51304 


•525 


1 •60968 


•587 


1-70745 


•649 


1-80780 


•402 


1^42092 


•464 


1-514.58 


•526 


1^61124 


•588 


1-70905 


•650 


1-80943 


•403 


1-42239 


•465 


1-51612 


•527 


1-61280 


•589 


1^71065 


•651 


1-81107 


•404 


1-42386 


•466 


1-51766 


.528 


1-61436 


•590 


1^71225 


•652 


1-81271 


•405 


1^42533 


•467 


1-51920 


.529 


1-61592 


•591 


1-71286 


•653 


1-81435 


•406 


1^42681 


•468 


1-52074 


.530 


1-61748 


•592 


1-71546 


•654 


1-81599 


•407 


1^42829 


•469 


1-52229 


.531 


1-61904 


•593 


1^71707 


•655 


1-81763 


•408 


r42977 


•470 


1-52384 


.532 


1-62060 


•594 


1^71868 


•656 


1-81928 


•409 


1-43125 


•471 


1-52539 


.533 


1-62216 


•595 


1^72029 


•657 


1-82091 


•410 


1-43273 


.472 


1-52691 


•534 


1-62372 


•596 


1-72190 


•658 


1-822-55 


•411 


1-43421 


•473 


1-52849 


.535 


1^62528 


•597 


1-723-50 


•659 


1-82419 


•412 


1-43569 


.474 


1-53004 


•536 


1^62684 


•598 


1-72511 


•660 


1-82-583 


•413 


1^43718 


.475 


1-53159 


•537 


1-62840 


•599 


1-72672 


.•661 


1-82747 


•414 


1^43867 


.476 


1-53314 


.538 


1-62996 


•600 


1^72833 


•662 


1-82911 


•415 


1^44016 


.477 


1-53469 


.539 


1-63152 


•601 


1-72994 


•663 


1-83075 


•416 


1^44165 


.478 


1-53625 


.540 


1-63309 


•602 


1-73155' 


•664 


1-83240 


•417 


1-44314 


.479 


1 •53781 


.541 


1-63465 


•603 


1-73316 


•665 


1-83404 


•418 


1 •44463 


.480 


1-53937 


.542 


1-63623 


•604 


1-73477 


-666 


1 •83568' 


•419 


1^44613 


.481 


1-54093 


.543 


1-63780 


-605 


1-73638 


-667 


1 •83733 


•420 


1-44763 


.482 


1-54249 


•544 


1-63937 


-606 


1^73799 


-668 


1 •83897 


•421 


1-44913 


.483 


1-54405 


•545 


1-64094 


-607 


1-73960 


•669 


1-84061 


•422 


1-45064 


.484 


1-54561 


.546 


1-64251 


•608 


1'74121 


•670 


1-84226 


•423 


1-45214 


.485 


1-54718 


.547 


1-64408 


-609 


1-74283 


•671 


1-84391 


•424 


1-45364 


.486 


1-54875 


.548 


1-64-565 


-610 


1-74444 


•672 


1-84-556 


•425 


1-45515 


.487 


1-55032 


.549 


1-64722 


•611 


1-74605 


•673 


1-84720 


•426 


1-45665 


.488 


1-55189 


•550 


1-64879 


-612 


1-74767 


•674 


1-84885 


Ji- 


1-45815 


.489 


1-55346 


•551 


1-65036 


•613 


1-74929 


•675 


1 -85050 


1-45966 


.490 


1-55503 


•552 


1-65193 


•614 


1-75091 


•676 


1-85215 


•429 


1-46167 


.491 


1-5-5660 


•553 


1-65350 


•615 


1-75252 


-677 


] -85379 


•430 


1-46268 


.492 


1-55817 


•554 


1-65507 


-616 


1-75414 


•678 


1-85544 


•431 


1-46419 


.493 


1-5-5974 


•555 


1-6-5665 


•617 


1-75-576 


-679 


1-85709 


•432 


1-46570 


.494 


1-56131 


•556 


1-65823 


-618 


1-75738 


•680 


1-85874 


•433 


1-46721 


.495 


1-56289 


•557 


1^6-5981 


•619 


1-75900 


•681 


1-86039 


•434 


1-46872 


•496 


1-56447 


•558 


1^66139 


•620 


1-76062 


-682 


1-86205 


•435 


1-47023 


.497 


1-56605 


•559 


1-66297 


•621 


1-76224 


•683 


1-86370 


•436 


1-47174 


.498 


1-56763 


•560 


1-66455 


•622 


1-76386 


•684 


1-86535 


•437 


1-47326 


.499 


1-56921 


•561 


1-66613 


•623 


1-76548 


•685 


1-86700 


•438 


1-47478 


.500 


1-57089 


-562 


1-66771 


•624 


1-76710 


-686 


1-86866 


•439 


1-47630 


.501 


1-57234 


-563 


1-66929 


•625 


1-76872 


-687 


1-87031 


•440 


1-47782 


.502 


1-57389 


•564 


1-67087 


•626 


1-77034 


-688 


1-87196 


•441 


1-47934 


.503 


1-57544 


-565 


1-67245 


•627 


1-77197 


•689 


1-87362 


•442 


1-48086 


•504 


1-57699 


-566 


1-67403 


•628 


1-773-59 


•690 


1-87527 


•443 


1-48238 


.505 


1-57854 


•567 


1-67561 


•629 


1-77521 


-691 


1-8769.S 


•444 


1-48391 


•506 


1-58009 


•568 


1-67719 


•630 


1^77684 


-692 


1-87859 


•445 


1-48544 


•507 


1.58164 


•569 


1-67877 


•631 


1^77847 


-693 


1-88024 


•446 

1 


1-48697 


608 


1-58319 


•570 


1-68036 


•632 


1^78009 


-694 


1-88190 



PROPORTIONS OF THE LENGTHS OF SEMI-ELLIPTIC ARCS. 



71 



Height 


Length of 


Height 


Length of 


Height 


Length of 


Height 


Lenirth of 


Height 1 Length of 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 


Arc. 


of Arc. 1 ^rc. 


•695 


1-88356 


•757 


1-98794 


•818 


2-09360 


-879 


2^20292 


-940 


12-31479 


•696 


1-88522 


-758 


1-98964 


•819 


2-09536 


-880 


2^20474 


-941 


1 2-31666 


•697 


1-88688 


•759 


1-99134 


-820 


2-09712 


-881 


2-20656 


-942 


2-318-52 


•698 


1-88854 


•760 


1-99305 


•821 


2-09888 


•882 


2-20839 


•943 


2-32038 


•699 


1-89020 


•761 


1-99476 


-822 


2-10065 


-883 


2-21022 


•944 


2-32224 


•700 


1^89186 


•762 


1-99647 


-823 


2-10242 


-884 


2-21205 


•945 


2-32411 


•701 


1-89352 


•763 


1-99818 


-824 


2-10419 


-885 


2-21388 


•946 


2-32598 


•702 


1-89519 


•764 


1-99989 


-825 


2-10596 


•886 


2-21571 


-947 


2-32785 


•703 


1-89685 


•765 


2-00160 


•826 


2-10773 


•887 


2-21754 


•948 


2-32972 


•704 


1-89851 


•766 


2-00331 


-827 


2-10950 


•888 


2-21937 


•949 


2-33160 


•705 


1-90017 


-767 


2-00502 


-828 


2-11127 


•889 


2-22120 


•950 


2-33348 


•706 


1-90184 


•768 


2-00673 


•829 


2-11304 


•890 


2-22.303 


•951 


2-33537 


•707 


1-90350 


•769 


2-00844 


-830 


2-11481 


•891 


2-22486 


•952 


2-33726 


•708 


1-90517 


•770 


2-01016 


•831 


2-11659 


•892 


2-22670 


•953 


2-33915 


•709 


1-90684 


•771 


2-01187 


-832 


2-11837 


•893 


2-22854 


•954 


2-34104 


•710 


1-90852 


-772 


2-01359 


-833 


2-12015 


•894 


2-23038 


•955 


2-34293 


•711 


1-91019 


•773 


2-01531 


•834 


2-12193 


•895 


2-23222 


•956 


2-34483 


•712 


1-91187 


•774 


2-01702 


•835 


2-12371 


•896 


2-23406 


•957 


2-34673 


•713 


1-91355 


•775 


2-01874 


•836 


2-12549 


•897 


2-23590 


•958 


2-34862 


•714 


1-91523 


-776 


2-02045 


-837 


2-12727 


•898 


2-23774 


•959 


2-3.5051 


•715 


1-91691 


•777 


2-02217 


•838 


2-12905 


•899 


2-23958 


•960 


2-35241 


•716 


1-91859 


•778 


2-02389 


•839 


2-13083 


•900 


2-24142 


•961 


2-35431 


•717 


1-92027 


•779 


2-02561 


-840 


2-13261 


•901 


2-24325 


•962 


2-35621 


•718 


1-92195 


•780 


2-02733 


-841 


2-13439 


•902 


2-24508 


•963 


2-35810 


•719 


1-92363 


•781 


2-02907 


•842 


2-13618 


•903 


2-24691 


•964 


2-36000 


•720 


1-92531 


•782 


2-03080 


•843 


2-13797 


•904 


2.24874 


•965 


2-36191 


•721 


1-92700 


•783 


2-03252 


-844 


2-13976 


•905 


2-25057 


•966 


2-36381 


•722 


1-92868 


•784 


2-03425 


•845 


2-14155 


•906 


2-25240 


•967 


2-36571 


-723 


1-93036 


•785 


2-03598 


-846 


2-14334 


•907 


2-25423 


•968 


2-36702 


•724 


1-93204 


•786 


2-03771 


-847 


2-14513 


•908 


2-25606 


•969 


2-36952 


•725 


1-93373 


-787 


2-03944 


•848 


2-14692 


•909 


2-25789 


•970 


2-37143 


•726 


1-93541 


•788 


2-04117 


•849 


2-14871 


•910 


2-25972 


•971 


2-373.34 


-727 


1-93710 


•789 


2-04290 


-850 


2-15050 


•911 


2-26155 


•972 


2-37-525 


•728 


1-93878 


•790 


2-04462 


•851 


2-15229 


•912 


2-26338 


•973 


2-37716 


•729 


1-94046 


-791 


2-04635 


•852 


2-15409 


•913 


2-26-521 


•974 


2-37908 


•730 


1-94215 


-792 


2-04809 


-853 


2-15589 


•914 


2-26704 


•975 


2-38100 


-731 


1-94383 


•793 


2-04983 


•854 


2-15770 


•915 


2-26888 


•976 


2-38291 


•732 


1-94552 


•794 


2-05157 


-855 


2-15950 


•916 


2-27071 


•977 


2-38482 


-733 


1-94721 


•795 


2-05331 


•856 


2-16130 


•917 


2-27254 


•978 


2-38673 


•734 


1-94890 


•796 


2-05505 


•857 


2-16309 


•918 


2-27437 


•979 


2-38864 


•735 


1-95059 


-797 


2-05679 


•858 


2-16489 


•919 


2-27620 


-980 


2-39055 


-736 


1-95228 


•798 


2-05853 


-859 


2-16668 


•920 


2-27803 


-981 


2-39247 


•737 


1-95397 


•799 


2-06027 


•860 


2-16848 


•921 


2-27987 


•982 


2-39489 


•738 


1-95566 


-800 


2-06202 


-861 


2-17028 


•922 


2-28170 


-983 


2-39631 


•739 


1-95735 


•801 


2-06377 


•862 


2-17209 


•923 


2-283.54 


•984 


2-39823 


-740 


1-95994 


•802 


2-06552 


•863 


2-17389 


•924 


2-28537 


•985 


2-40016 


•741 


1-96074 


-803 


2-06727 


•864 


2-17570 


•925 


2-28720 


•986 


2-40208 


•742 


1-96244 


-804 


2-06901 


•865 


2-17751 


-926 


2-28903 


•987 


2-40400 


•743 


1-96414 


•805 


2-07076 


•866 


2-17932 


-927 


2-29086 


-988 


2-40592 


•744 


1-96583 


■806 


2-07251 


•867 


2-18113 


•928 


2-29270 


•989 


2-40784 


•745 


1-96753 


•807 


2-07427 


•868 


2-18294 


•929 


2-29453 


•990 


2-40976 


•746 


1-96923 


•808 


2-07602 


•869 


2-18475 


•930 


2-29636 


•991 


2-41169 


•747 


1-97093 


-809 


2-07777 


-870 


2-18656 


•931 


2-29820 


•992 


2-41362 


•748 


1-97262 


-810 


2-D7953 


-871 


2-18837 


•932 


2-30004 


-993 


2-415-56 


•749 


1-97432 


•811 


2-08128 


•872 


2-19018 


-933 


2-30188 


-994 


2-41749 


•750 


1-97602 


•812 


2-08304 


•873 


2-19200 


•934 


2-30373 


•995 


2-41943 


-751 


1-97772 


-813 


2-08480 


•874 


2-19382 


•935 


2-30557 


•996 


2-42136 


-752 


1-97943 


•814 


2-08656 


•875 


2-19564 


•936 


2-30741 


•997 


2-42329 


•753 


1-98113 


-815 


2-08832 


•876 


2-19746 


-937 


2-30926! 


•998 


2-42522 


-754 


1 •98283 


-816 


2-09008 


•877 


2-19928 


-938 


2-31111 


•990 


2-42715 


•755 


1 -98453 


-817 


2-09198 


-878 


2-20110 


•939 


2-31295 


-1000 


2-42908 


•756 


1 -98023 

















72 THE PRACTICAL MODEL CALCULATOR. 

To find the length of an arc of a circle^ or the curve of a right 
semi-ellipse. 

Rule. — Divide tlie height by the base, and the quotient will be 
the height of an arc of which the base is unity. Seek, in the 
Table of Circular or of Semi-elliptical arcs, as the case may be, 
for a number corresponding to this quotient, and take the length 
of the arc from the next right-hand column. Multiply the number 
thus taken out by the base of the arc, and the product will be the 
length of the arc or curve required. 

In a Bridge, suppose the profiles of the arches are the arcs of 
circles ; the span of the middle arch is 240 feet and the height 24 feet ; 
required the length of the arc. 

24 -^ 240 = -100 ; and -100, by the Table, is 1-02645. 

Hence 1-02645 X 24 = 246-34800 feet, the length required. 

The profiles of the arches of a Bridge are all equal and similar 
semi-ellipses ; the span of each is 120 feet, and the rise 18 feet ; 
required the length of the curve. 

28 ^ 120 = -233 ; and -233 by the Table, is 1-19010. 

Hence 1-19010 x 120 = 142-81200 feet, the length required. 

In this example there is, in the division of 28 by 120, a remainder 
of 40, or one-third part of the divisor ; consequently, the answer, 
142-81200, is rather less than the truth. But this di0"erence, in 
even so large an arch, is little more than half an inch ; therefore, 
except where extreme accuracy is required, it is not worth com- 
puting. 

These Tables are equally useful in estimating works which may 
be carried into practice, and the quantity of work to be executed 
from drawings to a scale. 

As the Tables do not afford the means of finding the lengths of 
the curves of elliptical arcs which are less than half of the entire 
figure, the following geometrical method is given to supply the 
defect. 

Let the curve, of which the length is required to be found, be 
ABC. 



9 
Produce the height line Bt? to meet the centre of the curve 
in g. Draw the right line A^, and from the centre g, with the 
distance (/B describe an arc BA, meeting A^ in h. Bisect AA 
in z, and from the centre g with the radius gi describe the arc ik^ 
meeting d3 produced to k) then ik is half the arc ABC. 



TABLE OF RECIPROCALS OF NUMBERS. 



73 



A Table of the Reciproeals of Numbers ; or the Decimal Frac- 
tions corresponding to Vulgar Fractions of ivhich the Numera- 
tor is unity or 1. 

[In the following Tables, the Decimal fractions are Reciprocals 
of the Denominators of those opposite to them ; and their product 
is = unity. 

To find the Decimal corresponding to a fraction having a higher 
Numerator than 1, multiply the Decimal opposite to the given De- 
nominator, by the given Numerator. Thus, the Decimal corre- 
sponding to ^ being -015625, the Decimal to ^| will be •015625 X 
15 = -234375.] 



Fraction or 


Decimal or 


Fraetion or 


Decimal or 


Fraction or 


Decimal or 


Numb. 


Keciprocal. 


Numb. 


Eeciprocal. 


Numb. 


Keciprocal." 


1/2 


•5 


1/47 


•0212766 


1/92 


•010869565 


1/3 


•333333333 


1/48 


•020833333 


1/93 


•010752688 


1/4 


•25 


1/49 


•020408163 


1/94 


•010638298 


1/5 


•2 


1/50 


•02 


1/95 


•010526316 


1/6 


•166666667 


1/51 


•019607843 


1/96 


•010416667 


1/7 


•142857143 


1/52 


•019230769 


1/97 


•010309278 


1/8 


•125 


1/53 


•018867925 


1/98 


•010204082 


1/9 


•111111111 


1/54 


•018518519 


1/99 


•01010101 


1/10 


•1 


1/55 


•018181818 


1/100 


•01 


1/11 


•090909091 


1/56 


•017857143 


1/101 


•00990099 


1/12 


•083333333 


- 1/57 


•01754386 


1/102 


•009803922 


1/13 


•076923077 


1/58 


•017241379 


1/103 


•009708738 


1/14 


•071428571 


1/59 


•016949153 


1/104 


•009615385 


1/15 


•066666667 


1/60 


•016666667 


1/105 


•00952381 


1/16 


•0625 


1/61 


•016393443 


1/106 


•009433962 


1/17 


•058823529 


1/62 


•016129032 


1/107 


•009345794 


1/18 


•055555556 


1/63 


•015873016 


1/108 


•009259259 


1/19 


•052631579 


1/64 


•015625 


1/109 


•009174312 


1/20 


•05 


1/65 


•015384615 


1/110 


•009090909 


1/21 


•047619048 


1/66 


•015151515 


1/111 


•009009009 


1/22 


•045454545 


1/67 


•014925373 


1/112 


•008928571 


1/23 


•043478261 


1/68 


■014705882 


1/113 


•008849558 


1/24 


•041666667 


1/69 


•014492754 


1/114 


•00877193 


1/25 


•04 


1/70 


•014285714 


1/115 


•008695652 


1/26 


•038461538 


1/71 


•014084517 


1/116 


•00802069 


1/27 


•037037037 


1/72 


•013888889 


1/117 


•008547009 


1/28 


•035714286 


1/73 


•01369863 


1/118 


•008474576 


1/29 


•034482759 


1/74 


•013513514 


1/119 


•008403361 


1/30 


•033333333 


1/75 


•013333333 


1/120 


•008333333 


1/31 


•032258065 


1/76 


•013157895 


1/121 


■008264463 


1/32 


•03125 


1/77 


•012987013 


1/122 


•008196721 


1/33 


•030303030 


1/78 


•012820513 


1/123 


•008130081 


1/34 


•029411765 


1/79 


•012658228 


1/124 


•008064516 


1/35 


•028571429 


1/80 


•0125 


1/125 


•008 


1/36 


•027777778 


1/81 


•012345679 


1/126 


•007936508 


1/37 


•027027027 


1/82 


•012195122 


1/127 


•007874016 


1/38 


•026315789 


1/83 


•012048193 


1/128 


•0078125 


1/39 


•025641026 


1/84 


•011904762 


1/129 


•007751938 


1/40 


•025 


1/85 


•011764706 


1/130 


•007692308 


1/41 


•024390244 


1/86 


•011627907 


1/131 


•007633588 


1/42 


•023809524 


1/87 


•011491253 


1/132 


•007575758 


1/43 


•023255814 


1/88 


•011363636 


1/133 


•007518797 


1/44 


•022727273 


1/89 


•011235955 


1/134 


•007462687 


1/45 


•022222222 


1/90 


•011111111 


1/135 


•007407407 


1/46 


•021739ir 


1/91 


•010989011 


1/136 


•007352941 




G 











74 



THE PPtACTICAL MODEL CALCULATOR. 



Fractioa or 


Decimal or 


Fraction or 


Decimal or 


Fraction or 


Decimal or 


Numb. 


Eeciprocal. 


Numb. 


Eeciprocal. 


Numb. 


Eeciprocal. 


1/137 


•00729927 


1/198 


•005050505 


1,259 


•003861004 


1/138 


•007246377 


1;199 


•005025126 


i;260 


•003846164 


1/139 


•007194245 


1/200 


•006 


1261 


•003831418 


1/140 


•007142857 


1;201 


•004975124 


1,262 


•003816794 


1/141 


■007092199 


1,202 


•004950496 


1263 


•003802281 


1/142 


•007042254 


1/203 


•004926108 


1/264 


•003787879 


1/143 


•006993007 


1/204 


•004901961 


1/265 


•003773686 


1/144 


•006944444 


1/205 


•004878049 


1,266 


•003759398 


1/145 


•006896552 


1/206 


•004864369 


1/267 


•003745318 


1/146 


•006849315 


1/207 


•004830918 


1/268 


•003731343 


1/147 


•006802721 


1/208 


•004807692 


1/269 


•003717472 


1/148 


•006756767 


1/209 


•004784689 


1/270 


•003703704 


1/149 


•006711409 


1/210 


•004761905 


1,271 


•003690037 


1/150 


•006666667 


1/211 


•004739336 


1/272 


•003676471 


1/151 


•006622517 


1/212 


•004716981 


1/273 


•003663004 


1/152 


•006578947 


1/218 


•004694836 


1/274 


•003649635 


1/153 


•006535948 


1/214 


•004672897 


1/275 


•003636364 


1/154 


•006493506 


1/215 


•004651163 


1276 


•003623188 


1/155 


•006451613 


1/216 


•00462963 


1/277 


•003610108 


1/166 


•006410256 


1/217 


•004608296 


1/278 


•003597122 


1/157 


•006369427 


1/218 


•004687166 


1;279 


•003584229 


1/158 


•006329114 


1/219 


•00466621 


1/280 


•003571429 


1/159 


•006289308 


1/220 


•004545456 


1/281 


•003558719 


1/160 


•00625 


1/221 


•004524887 


1/282 


•003546099 


1/161 


•00621118 


1/222 


•004504605 


1/283 


•003533669 


1,162 


•00617284 


1/223 


•004484306 


1/284 


•003522127 


1/163 


•006134969 


1/224 


•004464286 


1/286 


•003508772 


1/164 


•006097561 


1/225 


•004444444 


1286 


•003496503 


1/165 


•006060606 


1/226 


•004424779 


1/287 


•003484321 


1/166 


•006024096 


1/227 


•004405286 


1/288 


•003472222 


1/167 


•005988024 


1/228 


•004386966 


1/289 


•003460208 


1/168 


•005952381 


1/229 


•004366812 


1/290 


•003448276 


1/169 


•00591716 


1/230 


•004347826 


1/291 


•003436426 


1/170 


•005882353 


1/231 


•004329004 


1/292 


•003424658 


1/171 


•005847953 


1/232 


•004310345 


1 293 


•003412969 


1/172 


•005813953 


1/233 


•004291846 


1294 


•003401361 


1/173 


•005780347 


1/234 


•004273504 


1/295 


•003389831 


1/174 


•005747126 


1/235 


■004255319 


l,/296 


•003378378 


1/175 


•005714286 


1/236 


•004237288 


1/297 


•003367003 


1176 


•005681818 


1/237 


•004219409 


1298 


•003366705 


1/177 


•005649718 


1/238 


•004201681 


1/299 


•003344482 


1/178 


•005617978 


1/239 


•0041841 


1300 


•003333333 


1/179 


•006586592 


1/240 


•004166667 


1/301 


•003322259 


1/180 


•005555556 


1/241 


•004149378 


1/302 


•003311258 


1/181 


•005524862 


1/242 


•004132231 


1/303 


•00330133 


1/182 


•005494505 


1/243 


•004116226 


1,'304 


•003289474 


1183 


•006464481 


1/244 


•004098361 


1/306 


•003278689 


1/184 


•005434783 


1/246 


•004081633 


1/306 


•003267974 


1/185 


•005406405 


1/246 


■004066041 


1,'307 


•003267329 


1/186 


•005376344 


1/247 


•004048583 


1 308 


•003246753 


1/187 


•006347594 


1/248 


•004032258 


1/309 


•003236246 


1/188 


•005319149 


1/249 


•004016064 


1/310 


•003225806 


1/189 


•005291005 


1/250 


•004 


1/311 


•003215434 


1/190 


•005263158 


1/251 


•003984064 


1/312 


•003205128 


1/191 


•006235602 


1/252 


•003968254 


1/313 


•003194888 


1/192 


•005208333 


1/263 


•003952569 


1,'314 


•003184713 


1/193 


•005181347 


1/254 


•003937008 


1/316 


•003174603 


1/194 


•005164639 


1/255 


•003921569 


1/316 


•003164657 


1/195 


•006128205 


1/256 


•00390625 


1/'317 


•003164674 


1/196 


•006102041 


1'257 


•003891051 


1/318 


•003144664 


1/197 


•005076142 


1258 


•003875969 


1/319 


•003134796 



TABLE OF REClrROCALS OF NUMBERS. 



75 



Fraction or 


Decimal or 


Fraction or 


Decimal or 


Fraction or 


Decimal or 


Numb. 


Reciprocal. 


Numb. 


Reciprocal. 


Numb. 


Reciprocal. 


1/320 


•003125 


1/381 


•002024672 


1/442 


•002262443 


1/321 


•003115265 


1/382 


•002617801 


1/443 


•002257336 


1/322 


•00310559 


1 383 


•002610966 


1/444 


•002252252 


1/323 


•003095975 


1/384 


•002604167 


1,445 


•002247191 


1/324 


•00308642 


1 385 


•002597403 


1/446 


•002242152 


1/325 


•003076923 


1/386 


•002590674 


1/447 


•002237136 


1/326 


•003067485 


1/387 


•002583979 


1/448 


■002232143 


1/327 


•003058104 


1/388 


•00257732 


1/449 


•002227171 


1/328 


•00304878 


1/389 


•002570694 


1/450 


•002222222 


1/329 


•003039514 


1/390 


•002564103 


1/451 


•002217295 


1/330 


•003030303 


1/391 


•002557545 


1/452 


•002212389 


1/331 


•003021148 


1/392 


•00255102 


1/453 


•002207506 


1/332 


•003012048 


1/393 


•002544529 


1/454 


•002202643 


1/333 


•003003003 


1/394 


•002538071 


1/455 


•002197802 


1/334 


•002994012 


1/395 


•002531646 


1/456 


•002192982 


1/335 


•002985075 


1/396 


•002525253 


1/457 


•002188184 


1,336 


•00297619 


1/397 


•002518892 


1/458 


•002183406 


1/337 


•002967359 


1/398 


•002512563 


1/459 


•002178649 


1/338 


•00295858 


1/399 


•002506266 


1/460 


•002173913 


1/339 


•002949853 


1/400 


•0025 


1/461 


•002169197 


1/340 


•002941176 


1/401 


•002493766 


1/462 


•002164502 


1/341 


•002932551 


1/402 


•002487562 


l/'463 


•002159827 


1/342 


•002923977 


1/403 


•00248139 


1/464 


•002155172 


1/343 


•002915452 


1/404 


•002475248 


1/465 


•002150538 


1/344 


•002906977 


1/405 


•002469136 


1/466 


•002145923 


1/345 


•002898551 


1/406 


•002463054 


1/467 


•002141328 


1/346 


•002890173 


l/'407 


•002457002 


1/468 


•002136752 


1/347 


•002881844 


1/408 


•00245098 


1/469 


■002132196 


1/348 


•002873563 


1/409 


•002444988 


1/470 


•00212766 


1/349 


•00286533 


1/410 


■002439024 


1/471 


•002123142 


1/350 


•002857143 


1/411 


•00243309 


1/472 


•002118644 


1/351 


•002849003 


1/412 


•002427184 


1/473 


•002114165 


1/352 


•002840909 


1/413 


•002421308 


1/474 


•002109705 


1/353 


•002832861 


1/414 


•002415459 


1/475 


■002105263 


1/354 


•002824859 


1/415 


•002409639 


1/476 


•00210084 


1/355 


•002816901 


1/416 


•002406846 


1/477 


•002096486 


1/356 


•002808989 


1/417 


•002398082 


1/478 


•00209205 


1/357 


•00280112 


1/418 


•002392344 


1/479 


■002087683 


1/358 


•002793296 


1/'419 


•002386635 


1/480 


•002083333 


1/359 


•002785515 


1/420 


•002380952 


1/481 


■002079002 


1/360 


•002777778 


1/'421 


•002375297 


1/482 


•002074689 


1/361 


•002770083 


1/422 


•002369608 


1/483 


■002070393 


1/362 


•002762431 


1/423 


•002364066 


1/484 


■002066116 


1/363 


•002754821 


1/424 


•002358491 


1/485 


•002061856 


1/364 


•002747235 


1/425 


•002352941 


1/486 


•002057613 


1/365 


•002739726 


1/426 


•002347418 


l/'487 


•002053388 


1/366 


•00273224 


1/427 


•00234192 


1/488 


•00204918 


1/367 


•002724796 


1/428 


•002336449 


1/489 


•00204499 


1/368 


■002717391 


1/429 


•002331002 


1/490 


■002040816 


1/369 


•002710027 


1/430 


•002325581 


1/491 


•00203666 


1/370 


•002702703 


1/431 


•002320186 


1/492 


•00203252 


1/371 


•002695418 


1/432 


•002314815 


1/493 


•002028398 


1/372 


•002688172 


1/433 


•002309469 


1/494 


•002024291 


1/373 


•002680965 


1/434 


•002304147 


1/495 


•002020202 


1/374 


•002673797 


1/435 


•002298851 


1/496 


•002016129 


1/375 


•002666667 


1/436 


•002293578 


1/497 


•002012072 


1/376 


•002659574 


1/437 


•00228833 


1/498 


•002008032 


1/377 


•00265252 


1/438 


•002283105 


1/499 


•002004008 


1/378 


•002645503 


1/439 


•002277904 


1/500 


•002 


1/379 


•002638521 


1/440 


•002272727 


1/501 


•001996008 


1/380 


•002631579 


1/441 


•002267574 


1/502 


•001992032 



76 



THE PRACTICAL MODEL CALCULATOR. 



Fraction or 


Decimal or 


Fraction or 


Decimal or 


Fraction or 


Decimal or 


Numb. 


Keciprocal. 


Numb. 


Reciprocal. 


Numb. 


Reciprocal. 


1,503 


•001988U72 


1,564 


•00177305 


1/625 


•0016 


1504 


•001984127 


1,565 


•001769912 


1;626 


•001597444 


1505 


•001980198 


1/566 


•001766784 


. 1/627 


•001594896 


1/506 


•001976285 


1/567 


•001763668 


1/628 


•001692357 


1/507 


•001972887 


1/568 


•001760563 


1/629 


•001589825 


1/508 


•001968504 


1/569 


•001757469 


1/630 


•001587302 


1;509 


•001964637 


1/570 


•001764386 


1/631 


•001584786 


1/510 


•001960784 


1/571 


•001751313 


1/632 


•001682278 


1/511 


•001956947 


1/572 


•001748252 


1,633 


•001579779 


1,512 


•001953125 


1/573 


•001745201 


1,634 


•001577287 


1/513 


•001949318 


1/574 


•00174216 


1,635 


•001574803 


1/514 


•001945525 


1/575 


•00173913 


1/636 


•001572327 


1/515 


•001941748 


1/576 


•001736111 


1/637 


•001569859 


1/516 


•001937984 


1/577 


•001733102 


1/638 


•001567398 


1/517 


•001934236 


1/578 


•001730104 


1/639 


•001564945 


1/518 


■001930502 


1/579 


•001727116 


1/640 


•0015625 


1/519 


•001926782 


1/680 


•001724138 


1,641 


•001560062 


1/520 


•001923077 


1/581 


•00172117 


1/642 


•001557632 


1/521 


•001919386 


1/582 


•001718213 


1/643 


•00155521 


1/522 


•001915709 


1/683 


•001715266 


1/644 


•001652795 


1/523 


•001912046 


1/684 


•001712329 


1,645 


•001560388 


1/524 


•001908397 


1/685 


•001709402 


1/646 


•001547988 


1/525 


•001904762 


1/586 


•001706485 


1/647 


•001545596 


1/626 


•001901141 


1/587 


•001703578 


1/648 


•00154321 


1/527 


•001897533 


1/588 


•00170068 


1/649 


•001540832 


1/528 


•001893939 


1/689 


•001697793 


1/650 


•001538462 


1/529 


•001890359 


1/590 


•001694915 


1,051 


•001536098 


1/530 


•001886792 


1/591 


•001692047 


1,652 


•001533742 


1/531 


•001883239 


1/692 


•001689189 


1/653 


•001531394 


1/532 


•001879699 


1,693 


•001686341 


1654 


•001529052 


1/533 


•001876173 


1,694 


•001683502 


l'655 


•001526718 


1/534 


•001872659 


1/596 


•001680672 


1/656 


•00152439 


1/535 


•001869159 


1/696 


•001677862 


1/667 


•00152207 


1/536 


•001865672 


1/697 


•001675042 


1,658 


•001519751 


1/537 


•001862197 


1/698 


•001672241 


1,659 


•001517451 


1/538 


•001858736 


1/699 


•001669449 


1/660 


•001515152 


1/539 


•001855288 


1/600 


•001666667 


1/661 


•001512859 


1/540 


•001851852 


1/601 


•001663894 


1/662 


•001510574 


1/541 


•001848429 


1/602 


■00166113 


1/663 


•001508296 


1/542 


•001845018 


1/603 


•001658375 


1/664 


•001506024 


1543 


•001841621' 


1/604 


•001655629 


1/665 


•001503759 


1/544 


•001838235 


1605 


•001652893 


1/666 


•001501602 


1/545 


•001834862 


1,606 


•001650165 


1/667 


•00149925 


1/546 


•001831502 


1/607 


•001647446 


1,'668 


•001497006 


1/547 


•001828154 


1/608 


•001644737 


1669 


•001494768 


1/548 


•001824818 


1/609 


•001642036 


1,670 


•001492537 


1/549 


•001821494 


1,610 


•001639344 


1/671 


•001490313 


1/550 


•001818182 


1/611 


•001636661 


1/672 


•001488095 


1/551 


•001814882 


1/612 


•001633987 


1/673 


•001485884 


1/552 


•001811594 


1/613 


•001631321 


1,674 


•00148368 


. 1/553 


•001808318 


1/614 


•001628664 


1,'675 


•001481481 


1/554 


•001805054 


1/615 


•001620016 


1/676 


•00147929 


1/555 


•001801802 


1/616 


•001623377 


1/677 


•001477105 


1/556 


•001798561 


1/617 


•001620746 


1,'678 


•001474926 


1/557 


•001795332 


1/618 


•001618123 


1,'679 


•001472754 


1/558 


•001792115 


1/619 


•001615509 


1/680 


•001470588 


1/559 


•001788909 


1/620 


•001612903 


1,'681 


•001468429 


1/560 


•001785714 


1/621 


•001610306 


1,'682 


•001466276 


1/561 


•001782531 


1/622 


•001607717 


1/683 


•001464129 ' 


1/562 


•001779359 


1/623 


•001605136 


1,'684 


•001461988 


1/563 


•001776199 


1/624 


•001602564 


1 685 


•001459854 



TABLE OF RECIPROCALS OF NUMBERS. 



77 



Fraction or 


Decimal or 


Fraction or 


Decimal or 


Fraction or 


1 

Decimal or 


Numb. 


Keciproeal. 


Numb. 


Ecciprocal. 


Numb. 


Keciproeal. 


1/686 


•001457726 


1/747 


•001338688 


1/808 


•001237624 


1/687 


•001455604 


1/748 


•001336898 


1/809 


•001236094 


1/688 


•001453488 


1/749 


•001335113 


1/810 


•001234568 


1/689 


•001451379 


1/750 


•001333333 


1/811 


•001233046 


1/690 


•001449275 


1/751 


•001331558 


1/812 


•001231527 


1/691 


•001447178 


1/752 


•001329787 


1/813 


•001230012 


1/692 


•001445087 


1/753 


•001328021 


1/814 


•001228501 


1/693 


•001443001 


1/754 


•00132626 


1/815 


•001226994 


1/694 


•001440922 


1/755 


•001324503 


1/816 


•001225499 


1/695 


•001438849 


1/756 


•001322751 


1/817 


•00122399 


1/696 


•001436782 


1/757 


•001321004 


1/818 


•001222494 


1/697 


•00143472 


1/758 


•001319261 


1/819 


•001221001 


1/698 


•001432665 


1/759 


•001317523 


1/820 


•001219512 


1/699 


•001430615 


1/760 


•001315789 


1/821 


•001218027 


1/700 


•001428571 


1/761 


•00131406 


1/822 


•001216545 


1/701 


•001426534 


1/762 


•001312336 


1/823 


•001215067 


1/702 


•001424501 


1/763 


•001310616 


1/824 


•001213592 


1/703 


•001422475 


1/764 


•001308901 


1/825 


•001212121 


1/704 


•001420455 


1/765 


•00130719 


l/'826 


. -001210654 


1/705 


•00141844 


1/766 


•001305483 


1/827 


•00120919 


1/706 


•001416431 


1/767 


•001303781 


1/828 


•001207729 


1/707 


•001414427 


1/768 


•001302083 


1/829 


•00120G273 


1/708 


•001412429 


1/769 


•00130039 


1/830 


•001204819 


1/709 


•001410437 


1/770 


•001298701 


1/831 


•001203369 


1/710 


•001408451 


1/771 


•001297017 


1/832 


•001201923 


1/711 


•00140647 


1/772 


•001295337 


1/833 


•00120048 


1/712 


•001404494 


1/773 


•001293661 


1/834 


•001199041 


1/713 


•001402525 


1/774 


•00129199 


1/835 


■•001197005 


1/714 


•00140056 


1/775 


•001290323 


1/836 


•001196172 


1/715 


•001398601 


1/776 


•00128866 


1/837 


•001194743 


1/716 


•001396648 


1/777 


•001287001 


1/838 


•001193317 


1/717 


•0013947 


1/778 


•001285347 


1/839 


•001191895 


1/718 


•001392758 


1/779 


•001283697 


1/840 


•001190476 


1/719 


•001390821 


1/780 


•001282051 


1/841 


•001189061 


1/720 


•001388889 


1/781 


•00128041 


1/842 


•001187648 


1/721 


•001386963 


1/782 


•001278772 


1/843 


•00118624 


1/722 


•001385042 


1/783 


•001277139 


1/844 


•001184834 


1/723 


•001383126 


1/784 


•00127551 


1/845 


•001183432 


1/724 


•001381215 


1/785 


•001273885 


1/846 


•001182033 


1/725 


•00137931 


1/786 


•001272265 


1/847 


•001180638 


1/726 


•00137741 


1/787 


•001270648 


1/848 


•001179245 


1/727 


•001375516 


1/788 


•001269036 


1/849 


•001177856 


1/728 


•001373626 


1/789 


•001267427 


1/850 


•001176471 


1/729 


•001371742 


1/790 


•001265823 


1/851 


•001175088 


1/730 


•001369863 


1/791 


•001264223 


1/952 


•001173709 


1/731 


•001367989 


1/792 


•001262626 


1/853 


•001172333 


1/732 


•00136612 


1/793 


•001261034 


1/854 


•00117096 


1/733 


•001364256 


1/794 


•001259446 


1/855 


•001109591 


1/734 


•001362398 


1/795 


•001257862 


1/856 


•001168224 


1/735 


•001360544 


1/796 


•001256281 


1/857 


•001166861 


1/736 


•001358696 


1/797 


•001254705 


1/858 


•001165501 


1/737 


•001356852 


1/798 


•001253133 


1/859 


•001164144 


1/738 


•001355014 


1/799 


•001251364 


1/860 


•001162791 


1/739 


•00135318 


1/800 


•00125 


1/861 


•00116144 


1/740 


•001351351 


1/801 


•001248439 


1/862 


•001160093 


1/741 


•001349528 


1/802 


•001246883 


1/803 


•001158749 


1/742 


•001347709 


1/803 


•00124533 


1/8G4 


•001157407 


1/743 


•001345895 


1/804 


•001243781 


1/865 


•001156069 


1/744 


•001344086 


1/805 


•001242236 


1/866 


•001154734 


1/745 


•001342282 


1/806 


•001240695 


1/867 


•001153403 


1/746 


•001340483 


1/807 


•001239157 


1/808 


•001152074 



g2 



78 



THE PRACTICAL MODEL CALCULATOR. 



Fraction or 


Decimal or 


Fraction or 


Decimal or 


Fraction or 


Decimal or 


Numb. 


Reciprocal. 


Kumb. 


Eeciprocal. 


Numb. 


Reciprocal. 


1/869 


•001150748 


1/913 


•00109529 


1/957 


•001044932 


1/870 


•001149425 


1/914 


•001094092 


1/958 


•001043841 


1/871 


•001148106 


1/915 


•001092896 


1/959 


•001042753 


1/872 


•001146789 


1/916 


•001091703 


1/960 


•001041667 


1/873 


•001145475 


1/917 


•001090513 


1/961 


•001040583 


1/874 


•001144165 


1/918 


•001089325 


1 962 


•001039501 


1/875 


•001142857 


1/919 


•001088139 


1/963 


•001038422 


1/876 


•001141553 


1/920 


•001086957 


1/964 


•001037344 


1/877 


•001140251 


1/921 


•001085776 


1/965 


•001036269 


1/878 


•001138952 


1/922 


•001084599 


1/966 


•001035197 


1/879 


•001137656 


1/923 


•001083423 


1/967 


•001034126 


1/880 


•001136364 


1/924 


•001082251 


1/968 


•001033058 


1/881 


•00113.5074 


1/925 


•001081081 


1/969 


•001031992 


1/882 


•001133787 


1/926 


•001079914 


1,970 


•001030928 


1/883 


•001132503 


1/927 


•001078749 


1/971 


•001029866 


1/884 


•001131222 


1/928 


•001077586 


1/972 


•001028807 


1/885 


•001129944 


1/929 


•001076426 


1/973 


•001027749 


1/886 


•001128668 


1/930 


•001075269 


1,974 


•001026694 


1/887 


•001127396 


1/931 


•001074114 


1,975 


•001025641 


1/888 


•001126126 


1/932 


•001072961 


1,976 


•00102459 


1/889 


•001124859 


1/933 


•001071811 


1/977 


•001023541 


1/890 


•001123596 


1/934 


•001070664 


1/978 


•001022495 


1/891 


•001122334 


1/935 


•001069519 


1/979 


•00102145 


1/892 


•001121076 


1/936 


•001068376 


1/980 


•001020408 


1/893 


•001119821 


1/937 


•001067236 


1/981 


•001019168 


1/894 


•001118568 


1938 


•001066098 


1/982 


•00101833 


1/895 


•001117818 


1;939 


•001064963 


1,983 


•001017294 


1/896 


•001110071 


1/940 


•00106383 


1/984 


•00101626 


1/897 


•001114827 


1/941 


•001062699 


1/985 


•001015228 


1,898 


•001113586 


1/942 


•001061571 


1,986 


•001014199 


1/899 


•001112347 


1/943 


•001060445 


1/987 


•001013171 


1/900 


•001111111 


1/944 


•001059322 


1,988 


•001012146 


1/901 


•001109878 


1/945 


•001058201 


1/989 


•001011122 


1/902 


•001108647 


1946 


•001057082 


1,990 


•001010101 


1/903 


•00110742 


1/947 


•001055966 


1/991 


•001009082 


1/904 


•001106195 


1/948 


•001054852 


1992 


•001008065 


1/905 


•001104972 


1/949 


•001053741 


1,993 


•001007049 


1/906 


•001103753 


1/950 


•001052632 


1,'994 


•001006036 


1/907 


•001102536 


1/951 


•001051525 


1/995 


•001005025 


1/908 


•001101322 


1/952 


•00105042 


1,996 


•001004016 


1/909 


•00110011 


1/953 


•001049318 


1/997 


•001003009 


1/910 


•001098901 


1/954 


•001048218 


1/998 


•001002004 


1/911 


•001091695 


1/955 


•00104712 


1/999 


•001001001 


1/912 


•001096491 


1/956 


•001046025 


1/1000 


•001 



Divide 80000 by 971. 

Bj the above Table we find tbatl divided by 971 gives -001029866, 
and -001029866 x 80000 = 82-38928. 
What is the sum of gf^ and gf-g ? 

5 X TSTT?. = -001132503 x 5 = 



2 x 



006 
1 

953 
5 



-001019318 

9 



•005662515 
X 2 = -002098636 



883 "^ 953 



= -007761141 



MENSURATION OF SOLIDS. 



79 



WEIGHTS AND VALUES IN DECIMAL PARTS. 



TROY WEIGHT. 


AVOIRDUPOIS 
WEIGHT. 


AVOIRDUPOIS 
WEIGHT. 


Dec. 


parts of a lb. 


Dec. 


parts of a evri. 


Dec 


parts of a lb. 


Ozs. 


Decimals. 


Qrs. 


Decimals. 


Ozs. 


Decimals. 


11 


•916666 


3 


•75 


15 


•9375 


10 


•833333 


2 


•5 


14 


•875 


9 


•75 


1 


•25 


13 


•8125 


8 


•666666 


lbs. 


Decimals. 


12 


•75 


7 


•583333 


27 


•241071 


11 


•6875 


6 


•5 


26 


•232142 


10 


•625 


5 


•416666 


25 


•223214 


9 


•5625 


4 


•333333 


24 


•214286 


8 


•5 


3 


•25 


23 


•205357 


7 


•4375 


2 


•166666 


22 


•196428 


6 


•375 


1 


•083333 


21 
20 


•187500 
•178572 


5 


•3125 


Dwts. 


Decimals. 


4 


•25 


19 


•079166 


19 


•169643 


3 


•1875 


18 


•075 


18 


•160714 


2 


•125 


17 


•070833 


17 


•151785 


1 


•0625 


16 


•066666 


16 


•142856 


Drs. 


Decimals. 


15 


•0625 


15 


•133928 


15 


•058593 


14 


•058333 


14 


•125 


14 


•054686 


13 


•054166 


13 


•116071 


13 


•050780 


12 


•05 


12 


•107143 


12 


•046874 


11 


•045833 


11 


•098214 


11 


•042968 


10 


•041666 


10 


•089286 


10 


•039062 


9 


•0375 


9 


•080357 


9 


•035156 


8 


•033333 


8 


•071428 


8 


•03125 


7 


•029166 


7 


•0625 


7 


•027343 


6 


•025 


6 


•053571 


6 


•023437 


5 


•020833 


5 


•044643 


5 


•019531 


4 


•016666 


4 


•035714 


4 


•015625 


3 


•0125 


3 


•026786 


3 


•011718 


2 


•008333 


2 


•017857 


2 


•007812 


1 


•004166 


1 


•008928 


1 


•003906 


Grs. 


Decimals. 


Ozs. 


Decimals. 






15 


•002604 


15 


•008370 


2J 


LONG 
EASURE. 


14 


•002430 


14 


•007812 






13 


•002257 


13 


•007254 


Dec. 


parts of a foot. 


12 


•002083 


12 


•006696 


Ins. 


Decimals. 


11 


•001910 


11 


•006138 


11 


•916666 


10 


•001736 


10 


•005580 


10 


•833333 


9 


•001562 


9 


•005022 


9 


•75 


8 


•001389 


8 


•001464 


8 


•666666 


7 


•001215 


7 


•003906 


7 


•583333 


6 


•001042 


6 


•003348 


6 


•5 


5 


•000868 


5 


•002790 


5 


•416666 


4 


•000694 


4 


•002232 


4 


•333333 


3 


•000521 


3 


•001674 


3 


•25 


2 


•000347 


2 


•001116 


2 


•166665 


1 


•000173 


1 


•000558 


1 


•083333 



To find the solidity of a euhe, the height of one 
of its sides being given. — Multiply the side of the 
cube by itself, and that product again by the side, 
and it will give the solidity required. 

The side AB, or BC, of the cube ABCDFGHE, 
is 25-5 : what is the solidity ? 

ITere AB' = (22-5)P = 25-5 X 25-5 x 25-5 = 
25-5 X 650-25 = 16581-375, content of the cube. 



F 




E 




\ 






G 




D 

n 




\ 


'n.^ 



80 



THE PKACTICAL MODEL CALCULATOE. 



D 1 

k- - ix 



To find the solidity of a parallelopipedon. 
— Multiply the length by the breadth, and 
that product again by the depth or altitude, 
and it will give the solidity required. 

Required the solidityof a parallelopipedon 
ABCDFEHG-, whose length AB is 8 feet, 
its breadth FD 4|- feet, and the depth or 
altitude AD 6f feet ? 

Sere AB X AD X FD = 8 X 6-75 x 4.5 = 54 x 4-5 = 243 solid 
feet, the contents of the imrallelopipedon. 

To find the solidity of a prism. — Multiply the area of the base 
into the perpendicular height of the prism, and the product will be 
the solidity. 

What is the solidity of the triangular prism ABCF 
ED, whose length AB is 10 feet, and either of the 
equal sides, BC, CD, or DB, of one of its equilateral ^. 



ends BCD, 2Jfeet? 






Here I X 2-5^ X v/3 = 
X v/3 = 1-5625 X 1-732 
base BCD. 

^ 2-5 + 2-5 + 2-5 


1 X 6-25 X v/3 = 1-5625 
= 2-70625 = area of the 

7'5 

-o- = 3-75 = i sum of 



the sides, BC, CD, DB, of the triangle CDB. 

And 3-75 - 2-5 = 1-25, .-. 1-25, 1-25 and 1-25 = 3 differences. 

Whence n/3-75 X 1-25 x 1-25 X 1-25 = v/3-75 X 1-25^ = 
^/7•32421875 = 2-7063 = area of the hase as hefore, 

And 2-7063 x 10 = 27*063 solid feet, the content of the prism 
required. 

To find the convex surface of a cylinder. — Multiply the peri- 
phery or circumference of the base, by the height of the cylinder, 
and the product will be the convex surface. 

jyf. .\g 

What is the convex surface of the right cylinder 
ABCD, whose length BC is 20 feet, and the diame- 
ter of its base AB 2 feet ? 



X 2 



20 



6-2832 = periphery of the 
- 125-6640 square feet, the 




Here 3-1416 
hase AB. 

And 6-2832 x 
convexity required 

To find the solidity of a cylinder. — Multiply the area of the 
base by the perpendicular height of the cylinder, and the product 
will be the solidity. 

What is the solidity of the cylinder ABCD, the diameter of 
»vhose base AB is 30 inches, and the height BC 50 inches. 

Here '7854 X 30^ = -7854 x 900 = 706-86 = area of the hase AB, 

And 706-86 x 50 = 35343 cuUc inches; or~^ = 20-4531 



solid feet. 



1728 



MENSURATION OF SOLIDS. 



81 



obliquely through the opposite 





The four following cases contain all the rules for finding the su- 
perficies and solidities of cylindrical ungulas. f 

When the section is parallel to the axis of the cylinder. ^ 
Rule. — Multiply the length of the arc line of the base 

by the height of the cylinder, and the product will be 

the curve surface. 

Multiply the area of the base by the height of the a 

cylinder, and the product will be the solidity. 

When the 
sides of the cylinder. 

Rule. — Multiply the circumference of the base of the 
cylinder by half the sum of the greatest and least lengths e 
of the ungula, and the product will be the curve surface. 

Multiply the area of the base of the cylinder by half 
the sum of the greatest and least lengths of the ungula, and the 
product will be the solidity. 

When the section passes through the base of the cylin- j 
der, and one of its sides. 

Rule. — Multiply the sine of half the arc of the base 
by the diameter of the cylinder, and from this product 
subtract the product of the arc and cosine. 

Multiply the diiference thus found, by the quotient of b 
the height divided by the versed sine, and the product 
will be the curve surface. 

From f of the cube of the right sine of half the arc of the base, 
subtract the product of the area of the base and the cosine of the 
said half arc. 

Multiply the difi'erence, thus found, by the quotient arising from 
the height divided by the versed sine, and the product will be the 
solidity. 

When the section passes obliquely through both ends 
of the cylinder. 

Rule. — Conceive the section to be continued, till it 
meets the side of the cylinder produced ; then say, as 
the difi'erence of the versed sines of half the arcs of the 
two ends of the ungula is to the versed sine of half the 
arc of the less end, so is the height of the cylinder to 
the part of the side produced. 

Find the surface of each of the ungulas, thus formed, and their 
difi'erence will be the surface. 

In like manner find the solidities of each of the ungulas, and 
their difi'erence will be the solidity. 

To find the convex surface of a right cone. — Multiply the circum- 
ference of the base by the slant height, or the length of the side 
of the cone, and half the product will be the surface required. 

The diameter of the base AB is 3 feet, and the slant height 
AC or BC 15 feet; required the convex surface of the cone 
ACB. 




82 



THE PRACTICAL MODEL CALCULATOE. 



Jffere 3-1416 x 3 = 9-4248 = circumference of the hase AB. 

^ , 9-4248 X 15 141-3720 ^^ ^^^ 

And ^ = ^ = 70-686 square feet, the convex 

surface required. 

To find the convex surface of the frustum of a right cone. — Mul- 
tiply the sum of the perimeters of the two ends, by the slant height 
of the frustum, and half the product will be the surface required. 

In the frustum ABDE, the circumferences of 
the two ends AB and DE are 22-5 and 15-75 
respectively, and the slant height BD is 26 ; what 
is the convex surface ? Er- ::ad 

^^ (22-5 + 15-75) X 26 _— -— — 

Here ^ ^ — " = ^^'^ + ^^'^^ 

X 13 = 38-25 X 13 = 497*25 = convex sur- 

'^ace. At- ^B 

To find the solidity of a cone or pyramid. — Multiply the area of 
the base by one-third of the perpendicular height of the cone or 
pyramid, and the product will be the solidity. 




Required the solidity of the cone ACB, whose 
diameter AB is 20, and its perpendicular height 
CS24. 

Here '7854 X 20^ = -7854 X 400 = 314-16 
= area of the hase AB. 

And 314-16 X y = 314-16 x 8 = 2513-28 
= solidity required. 



Required the solidity of the hexagonal pyra- 
mid ECBD, each of the equal sides of its base 
being 40, and the perpendicular height CS 60. 

Here 2-598076 [multiplier when the side is 1) 
X 40^ = 2-598076 x 1600 = 4156-9216 = area 
of the hase. 

60 

And 4156-9216 x -tt = 4156-9216 x 20 = 



83138-432 solidity. 




To find the solidity of a frustum of a cone or pyramid. — For 
the frustum of a cone, the diameters or circumferences of the two 
ends, and the height being given. 

Add together the square of the diameter of the greater end, the 
square of the diameter of the less end, and the product of the two 



MEXSURATION OF SOLIDS. 83 

diameters ; multiply the sum by -7854, and the product by the 
height ; ^ of the last product will be the solidity. Or, 

Add together the square of the circumference of the greater 
end, the square of the circumference of the less end, and the pro- 
duct of the two circumferences; multiply the sum by '07958, and 
the product by the height ; J of the last product will be the solidity. 

Foi' the frustum of a2)yramid tvhose sides are regular 'polygons. — 
Add together the square of a side of the greater end, the square 
of a side of the less end, and the product of these two sides ; mul- 
tiply the sum by the proper number in the Table of Superficies, and 
the product by the height ; \ of the last product will be the solidity. 

Wlien-tlie ends of tlie pyramids are not regular polygons. — Add 
together the areas of the two ends and the square root of their 
product ; multiply the sum by the height, and \ of the product 
will be the solidity. 

What is the solidity of the frustum of the cone e^ 
EABD, the diameter of whose greater end AB is / 
5 feet, that of the less end ED, 3 feet, and the / 
perpendicular height Ss, 9 feet ? / 

(5^ + 3^ + TyTZ) X -7854 x 9 _ 346-3614 _ / ^ 

3 ~ 3 ~ A^ 
115-4538 solid feet.^ the content of the frustum. 

What is the solidity of the frustum eEDB5 of a e^ 

hexagonal pyramid, the side ED of whose greater / / 

end is 4 feet, that eb of the less end 3 feet, and / /■ 

the height Ss, 9 feet ? / /; 

(4^ + 3^ + 4 X 3) X 2-598076 x 9 865-159308 / | 

= 288-386436 solid feet, the solidity required. ^ 

The following cases contain all the rules for finding the superficies 
and solidities of conical ungulas. 

When the section passes through the op)]posite extremities of the 
ends of the frustum. 

Let D = AB the diameter of the greater end ; ^i^^f* 
d = CD, the diameter of the less end ; h = perpen- / \ \\\ 
dicular height of the frustum, and n == -7854. / VyA 

^. d^-d^/J)d nWi ,.,. , / V\\ 

Then — ^^ _ , — X —^ = solidity of the greater ^k::::;;;j<^3 

elliptic ungula ADB. 
Bx/Dd-d' ndh 
— ij _ J X ~o~ ~ solidity of the less ungula ACD. 

(d'^-^V nh ^.^ 
jy _ ^ ^ "S" ~ difference of these hoofs. 

^nd jj-ZT^ v/4/i2 + (D - d^ X (D2 ^ VDd = curve 

surface of ADB. 




84 THE PRACTICAL MODEL CALCULATOR. 

IVIien the section cuts off parts of the base, and makes the angle 
DrB less than the angle CAB. 

Let S = tabular segment, whose v ersed sine is c^ 
Br-T-D; s — tab. seg. whose versed sine is Br — (D — d) 
-T- dy and the other letters as above. 

Br Br / _A 

The (S X D^ - s X c^^ X ==== V- — ff=y 

^ Br — D — c^ Br— D— rf 

X p^_^ = solidity of the elliptic hoof EFBD. 

And p-3^ v/4A^ + (D - (^f x(seg. FBE-^.X ^ d - Lr 

Br ^ d — Ar^ 
X V n _ K X seg. of the circle AB, whose height is D X — ^ ) 

= convex surface of EFBD. 

When the section is parallel to one of the sides of the frustum. 
Let A = area of the base FBE, and the other let- ^^^\ 

ters as before. /f\ \ 

Then ( -^ ^ ^ - ^d v/(B — d) x d) X J/i = solidity / \\\ \ 
of the parabolic hoof EFBD. ^(^g^B 

And ^^Td %/4A' x (D - df x (seg. FBE - 1 D^ ^ 

X s/d xD — d) = convex surface of EFBD. 

When the section cuts off part of the base, and makes the angle 
DrB greater than the angle CAB. 

Let the area of the hyperbolic section EDF = A, ^^ — ^^ 

and the area of the circular seg. EBF = a. [^\\ 

Vi , -r. . ^ xEr .,. . yl \\ \ 

Then ^ _ ^ X (a x D — A X ^^ ) = solidity of / \ \ \ 

the hyperbolic ungula EFBD. J^:}v'.'.y{::\^ 

1 ^Y^^-^ 

And p_^ X ^/4A2 + (D - cTf X (cir. seg. EBF - 

d"^ Br-lD-cZ) Br ^ p -n-n-nrx 

:pr-, X — — - — ' \/— - = curve surface of EiBD. 

D^ Br - D - (i Br - t^ - D 

dxQ;r 
The transverse diameter of the hyp. seg. = t-^ _ ^ _ -n and the 

Br 

conjugate = d V j. _ ^ _ p ? ^^s^om which its area may be found 

by the former rules. " 

To find the solidity of a cuneus or ivedge. — Add tAvice the length 
of the base to the length of the edge, and reserve the number. 

Multiply the height of the wedge by the breadth of the base, 
and this product by the reserved number ; \ of the last product 
will be the solidity. 



MENSURATION OF SOLIDS. 



85 



How many solid feet are there in a wedge, 
"wliose base is 5 feet 4 inches long, and 9 inches 
broad, the length of the edge being 3 feet 6 inches, 
and the perpendicular height 2 feet 4 inches ? 



(64 X 2 + 42) X 28 X 9 (128 + 42) x 28 x 



E F 




\ 

\ 


/" 


;p 



= 170 X 14 X 3 = 7140 solid 



Here 

170 X 28 X 9 170 X 28 X 3 

6 ~ 2 

incJies. 

And 7140 -^ 1728 = 4-1319 solid feet, the content. 

To find the solidity of a prismoid. — To the sum of the areas of 
the two ends add four times the area of a section parallel to and 
equally distant from both ends, and this last sum multiplied by \ 
of the height will give the solidity. 

The length of the middle rectangle is equal to half the sum of 
the lengths of the rectangles of the two ends, and its breadth equal 
to half the sum of the breadths of those rectangles. 

What is the solidity of a rectangle prismoid, 
the length and breadth of one end being 14 and 
12 inches, and the corresponding sides of the other 
6 and 4 inches, and the perpendicular 30J feet. 




Eere 14 X 12 + 6 X 4 = 168 + 24 = 192 = d^ 
mm of the area of the two ends, 
14 + 6 20 
Also — 2 — ^ 2" ~ -^^ ~ length of the middle rectangle. 

^ 12 + 4 16 ^ 

And — ^ = — = 8 = breadth of the middle rectangle. 



80 X 4 = 320 = 4 times 



area of 



2 ~ 2 
Whence 10 X 8 X 4 = 
the middle rectangle. 

366 
Or (320 -f 192) x -^ = 512 x 61 = 31232 solid inches. 

And 31232 -^ 1728 = 18-074 solid feet, the content. 

To find the convex surface of a s;phere. — Multiply the diameter 
of the sphere by its circumference, and the product will be the 
convex superficies required. 

The curve surface of any zone or segment will also be found by 
multiplying its height by the whole circumference of the sphere. 

D 

What is the convex superficies of a globe 
BCGr whose diameter BG is 17 inches ? 

Here 3-1416 _X 17 X 17 = 53-4072 x 17 = 
907*9224 square inches. ^ 

And 907-9224 -v- 144 = 6-305 square feet. 



H 




86 



THE PRACTICAL MODEL CALCULATOR. 





To find the solidity of a sphere or glohe. — Multiply tlie cube of 
the diameter by -5236, and the product will be the solidity. 

What is the solidity of the sphere AEBC, 
whose diameter AB is 17 inches ? 

Here IV X -5236 = 17 x 17 X 17 X -5236 = 
289 X 17 X'5236 = 4913 x -5236 = 2572-4468 

And2572'44:6S -^ 1728 = 1-48868 solid feet. 

To find the solidity of the segment of a sphere. — To three times 
the square of the radius of its base add the square of its height, and 
this sum multiplied by the height, and the product again by -5236, 
will give the solidity. Or, 

From three times the diameter of the sphere subtract twice the 
height of the segment, multiply by the square of the height, and 
that product by -5236 ; the last product will be the solidity. 

The radius Qn of the base of the segment 
CAD is 7 inches, and the height An 4 inches ; 
what is the solidity ? 

Eere (7' X 3 + 4^) x 4 X -5236 = (49x3+4^) 
x4 X -5236 = (147 + 4^) x4x -5236 = (147+16) \ I 

X 4 x -5236 = 163 X 4 X -5236 = 652 x -5236 \ I / 
.= 341-3872 solid inches. ""--._[--'^' 

B 

To find the solidity of a frustum or zone of a sphere. — To the 
sum of the squares of the radii of the two ends, add one-third of 
the square of their distance, or of the breadth of the zone, and 
this sum multiplied by the said breadth, and the product again by 
1-5708, will give the solidity. 

What is the solid content of the zone ABCD, 
whose greater diameter AB is 20 inches, the ^' 
less diameter CD 15 inches, and the distance 
nm of the two ends 10 inches ? 

Eere (10^ + 7-5^ + ^) X 10 X 1-5708 = 

(100 + 56-25 + 33-33) x 10 x 1-5708 = 189-58 

X 10 X 1-5708 = 1895-8 x 1-5708 = 2977-92264 solid inches. 

To find the solidity of a spheroid. — Multiply the square of the 
revolving axe by the fixed axe, and this product again by -5236, 
and it will give the solidity required. 

'5236 is = 1 of 3-1416. 

In the prolate spheroid ABCD, the ^ 

transverse, or fixed axe AC is 90, and 
the conjugate or revolving axe DB is 70 ; 
what is the solidity ? 

Eere DB^ X AC X -5236 = 70^ x 90 
X -5236 = 4900 x 90 x -5236 = 441000 
X -5236 = 230907-6 = solidity reciuired. 



>— ^^^^^^^ 


K:::-^mz:;^ 




MENSURATION OF SOLIDS. 



87 




To find the content of the middle frustum of a spheroid, its 
length, the middle diameter, and that of either of the ends, being 
given, when the ends are circular or parallel to the revolving axis. — 
To twice the square of the middle diameter add the square of the 
diameter of either of the ends, and this sum multiplied by the length 
of the frustum, and the product again by -2618, will give the solidity. 

Where -2618 = ^^ of 3-1416. 

In the middle frustum of a spheroid 
EFG-H, the middle diameter DB is 
50 inches, and that of either of the 
ends EF or GH is 40 inches, and its ^< 
length nm 18 inches; what is its soli- 
dity ? 

Here (50^ X 2 + 40^) x 18 X -2618 
= (2500 X 2 + 1600) X 18 X -2618 = (5000 + 1600) x 18 x 
•2618 = 6600 X 18 X -2618 = 118800 x -2613 = 31101-84 cubic 
iyiches. 

When the ends are elliptical or perpendicular to the revolving 
axis. — Multiply twice the transverse diameter of the middle sec- 
tion by its conjugate diameter, and to this product add the product 
of the transverse and conjugate diameters of either of the ends. 

Multiply the sum thus found by the distance of the ends or 
the height of the frustum, and the product again by *2618, and it 
will give the solidity required. 

In the middle frustum ABCD of an oblate ^ 

spheroid, the diameters of the middle section 
EF are 50 and 30, those of the end AD 40 
and 24, and its height ne 18; what is the e( 
solidity ? 

Here (50 X 2 X 30 -f 40 x 24) x 18 X -2618 
= (3000 + 960) X 18 X -2618 = 3960 x 18 x 
•2618 = 71280 X -2618 = 18661-104 = the solidity. 

To find the soUdity of the segment of a spheroid, ivhen the base 
is parallel to the revolving axis. — Divide the square of the revolv- 
ing axis by the square of the fixed axe, and multiply the quotient 
by the difference between three times the fixed axe and twice the 
height of the segment. 

Multiply the product thus found by the square of the height of 
the segment, and this product again by -5236, and it will give the 
solidity required. 

In the prolate spheroid DEFD, the trans- 
verse axis 2 DO is 100, the conjugate AC 60, 
and the height Dn of the segment EDF 10 ; 
what is the solidity ? 

602 

Here (^-^^ X 300 - 20) x 10^ x -5236 = 

-L^^ A 

•36 X 280 X 102 X .5236 = 10O-8O x 100 x -5236 = 10080 x 
•5236 = 5277-888 = the solidity. 




D 

:::llll — - — ~ "3 . F 



THE PRACTICAL MODEL CALCULATOR. 




When the base is perpendieular to the revolving axis. — Divide the 
fixed axe by the revolving axe, and multiply the quotient by the 
difference between three times the revolving axe and twice the 
height of the segment. 

Multiply the product thus found by the square of the height of 
the segment, and this product again by -5236, and it will give the 
solidity required. 

In the prolate spheroid aE6F, the trans- 
verse axe EF is 100, the conjugate ah 60, and 
the height an of the segment <xAD 12 ; what 
is the solidity ? ^'r>-I.V.„/i ...--'7 

Here 156 (= diff. of Sab and 2an) X If ^x^ I ^.^' 
(= EF ^ a5 X 144 (= square of an) X -5236 '"^"f ^ 

156 X 5 
= g X 144 X -5236 = 52 x 5 x 144 x -5236 = 260 x 

144 X -5236 = 37440 x -5236 = 19603-584 = the soliditij. 

To find the solidity of a parabolic conoid. — Multiply the area of 
the base by half the altitude, and the product will be the content. 

What is the solidity of the paraboloid ADB, 
whose height Dm is 84, and the diameter BA 
of its circular base 48 ? 

Here 48^ X -7854 x 42 (= | Dm) = 2304 x 
•7854 X 42 = 1809-5616 x 42 = 76001-5872 
= the soliditij. 

To find the solidity of the frustum of a paraboloid, when its ends 
are perpendicular to the axe of the solid. — Multiply the sum of the 
squares of the diameters of the two ends by the height of the frus- 
tum, and the product again by -3927, and it will give the solidity. 

Required the solidity of the parabolic frus- /'' "P'~~n 

tum ABCcZ, the diameter AB of the greater end 
being 58, that of the less end dc 30, and the 
height no 18. 

Here (58^ -f 30^) x 18 X -3927 = (3364 -f 
900) X 18 X -3927 = 4264 x 18 x -3927 = 
76752 X -3927 = 30140-5104 = the solidity. 

To find the solidity of an hyperboloid. — To the square of the 
radius of the base add the square of the middle diameter between 
the base and the vertex, and this sum multiplied by the altitude, 
and the product again by -5236 will give the solidity. 

In the hyperboloid ACB, the altitude Cr 
is 10, the radius Ar of the base 12, and the mid- 
dle diameter nm 15-8745; what is the solidity? 

Here 15-8745^ -f 12^ X 10 x -5236 = 
251-99975 + 144 x 10 x -5236 = 395-99975 x 
10 X -5236 = 3959-9975 x -5236 = 2073-454691 
= the solidity. 




I n \ 

.--'^"2_ — --" 

:r '"o " ^-' 




MENSURATION OF SOLIDS. 



89 



To find the solidity of the frustum of an hyperbolic conoid. — Add 
together the squares of the greatest and least semi-diameters, and 
the square of the whole diameter in the middle; then this sum being 
multiplied by the altitude, and the product again by •5236, -will 
give the solidity. 

In the hyperbolic frustum ADCB, the length ^,,?^ 

rs is 20, the diameter AB of the greater end 32, 
that DC of the less end 24, and the middle dia- 
meter nm 28*1708; required the solidity. 

Here (16^ + 12^ + 28-17082) x 20 X -52359 
= (256 + 144 + 793-5939) x 20 x -52359 = 
1193-5939 X 20 X -52359 = 23871-878 x -52359 
= 12499-07660202 = solidity. 

To find the solidity of a tetraedron. — Multiply ^^ 
of the cube of the linear side by the square root of 
2, and the product will be the solidity. 

The linear side of a tetraedron ABCw is 4 ; what 

is the solidity ? 

4^ 4x4x4 _ 4x4 o _ ^^ 

X s/ 2 — :i7) X -\/ 2 — Q X v 2 — Q 





12 



v/2 = yx 



12 
1-414 



22-624 



3 
7-5413 



solidity. 



To find the solidity of an octaedron. — Multiply J of the cube 
of the linear side by the square root of 2, and the product will be 
the solidity. 



What is the solidity of the octaedron BGAD, 
whose linear side is 4 ? 

43 64 

-gXv/2 = yXv/2 = 21-333, x v/2 = 

21-333 X 1-414 = 30-16486 = solidity. 



To find the solidity of a dodecaedron. — To 21 times the square 
root of 5 add 47, and divide the sum by 40 : then the square root 
of the quotient being multiplied by five times the cube of the linear 
side will give the solidity. 

The linear side of the dodecaedron ABODE 
is 3 ; what is the solidity ? 

21 n/ 5 + 47 ^^ , 21x2-23606+47 
\/ 7a X 27 X 5= V — 




40 
X 27 X 5 = v/ 



46-95726 + 47 
40 



40 



X 135 = 206-901 




solidity. 

To find the solidity of an icosaedron. — To three times the square 
root of 5 add 7, and divide the sum by 2 ; then the square root of 



h2 



90 



THE PRACTICAL MODEL CALCULATOR. 



this quotient being multiiDlied by | of the cube of tbe linear side 
"will give the solidity. 

7 4- 3 -v/ 5 
That is I S^ X \/ ( o ) = solidity when S is = to the 

linear side. 

The linear side of the icosaedron ABCDEF 
is 3 ; what is the solidity ? 

3v/5 +7 5x3^ 3 X 2-23606 + 7 c 

V 2 ^ 6 ~ ^ 2 

5 X 27 6-70818 + 7 5x9 

X— g— = ^/ 2 '^ 

13-70818 45 

^/ 2 >^T^ v/6-85409 X 22-5 

X 22-5 = 58-9056 = solidity. 

The superficies and solidity of any of the five regular bodies may 
be found as follows ; 

Rule 1. Multiply the tabular area by the square of the linear 
edge, and the product will be the superficies. 

2. Multiply the tabular solidity by the cube of the linear edge, 
and the product will be the solidity. 





Surfaces and Solidities of tJie Regular Bodies. 



No. of 

Sides. 


Names. 


Surfaces. 


Solidities. 


4 


Tetraedron 


1.73205 


0.11785 


6 


Hexaedron 


6.00000 


1.00000 


8 


Octaedron 


3.46410 


0.47140 


12 


Dodecaedron 


20.64578 


7.66312 


20 


Icosaedron 


8.66025 


2.18169 



To find the convex superficies of a cylindrie ring. — To the thick- 
ness of the rino' add the inner diameter, and this sum beino; multi- 
plied by the thickness, and the product again by 9.8696, will give 
the superficies. 

The thickness of Kc of a cylindrie ring is 3 
inches, and the inner diameter cd 12 inches ; 
what is the convex superficies ? 

lF+~3 X 3 X 9-8696 = 15 x 3 x 9-8696 
= 45 X 9-8696 = 444-132 = superficies. 

To find the solidity of a cylindrie ring. — To the thickness of the 
ring add the inner diameter, and this sum being multiplied by the 
square of half the thickness, and the product again by 9-8696, 
will give the solidity. 




MENSURATION OF SOLIDS. 91 

What is the solidity of an anchor ring, whose inner diameter is 
inches. 



8 + 3 X ||2 X 9-8696 = 11 x 1-5^ x 9-8693 = 11 x 2-25 x 
9-8696 = 24-75 x 9-8696 = 2M-2T26 = solidity. 

The inner diameter AB of the cylindric ring / 

cc?e/ equals 18 feet, and the sectional diameter z;^^^^^^ 
cK or Be equals 9 inches ; required the convex A'^^^^^^^k. 
surface and solidity of the ring. I M y|| 

18 feet X 12 = 216 inches, and 216 + 9 WI iW 

X 9 X 9-8696 = 19985-94 square inches. \^^ yW 

216+ 9 X 92 X 2-4674 = 44968-365 cubic ^^^^^ 
inches. d 

In the formation of a hoop or ring of wrought iron, it is found 
in practice that in bending the iron, the side or edge which forms 
the .interior diameter of the hoop is upset or shortened, while at 
the same time the exterior diameter is drawn or lengthened ; there- 
fore, the proper diameter by which to determine the length of the 
iron in an unbent state, is the distance from centre to centre of the 
iron of which the hoop is composed : hence the rule to determine 
the length of the iron. If it is the interior diameter of the hoop 
that is given, add the thickness of the iron ; but if the exterior di- 
ameter, subtract from the given diameter the thickness of the iron, 
multiply the sum or remainder by 3-1416, and the product is the 
length of the iron, in equal terms of unity. 

Supposing the interior diameter of a hoop to be 32 inches, and 
the thickness of the iron 1 J, what must be the proper length of the 
iron, independent of any allowance for shutting? 

32 + 1-25 = 33-25 x 3-1416 = 104-458 inches. 
But the same is obtained simply by inspection in the Table of Cir- 
cumferences. 
Thus, 33-25 = 2 feet 9| in., opposite to w^hich is 8 feet 8^ inches. 

Again, let it be required to form a hoop of iron J inch in thick- 
ness, and 16J inches outside diameter. 

16-5 - -875 = 15-625, or 1 foot 3| inches ; 
opposite to which, in the Table of Circumferences, is 4 feet 1 inch, 
independent of any allowance for shutting. 

The length for angle iron, of which to form a ring of a given di- 
ameter, varies according to the strength of the iron at the root ; 
and the rule is, for a ring with the flange outside, add to its required 
interior diameter, twice the extreme strength of the iron at the 
root ; or, for a ring with the flange inside, sub- c d c d 

tract twice the extreme strength ; and the sum or [7^ *^ 

remainder is the diameter by which to determine ri' bi 

the length of the angle iron. Thus, suppose two i i 

angle iron rings similar to the following be re- !^ -^1 

quired, the exterior diameter AB, and interior ^ IV-, 

diameter CD, each to be 1 foot lOJ inches, and c d c d 

the extreme strength of the iron at the root cd, cd, &c, J of an inch ; 



92 



THE PRACTICAL MODEL CALCULATOR. 



twice 1^ = If, 



and 1 ft. 10| in. + If = 2 ft. | in., opposite to 
in the Table of Circumferences, is 6 ft. 4 J in., the length of 



■which 

the iron for CD ; and 1 ft. lOJ in. - If = 1 ft. 8f 

to which is 5 ft. 5i in., the length of the iron for AB. 



m., opposite 



in addition to the length 



But observe, as before, that the necessary allowance for shutting 
must be added to the length of the iron, 
as expressed by the Table. 

Required the capacity in gallons of a 
locomotive engine tender tank, 2 feet 8 
inches in depth, and its superficial di- 
mensions the following, with reference 
to the annexed plan : 




Length, or dist. between A and B = 10 ft. 


2f in 


or, 


— - y 
122-75 in. 


Breadth " C and D = 6 


7* 




79-5' 


Length '' i and ^ = 3 


lOf 




46-75 


Mean breadth of coke- \ , ^ _ g 

space or - J 
Diameter of circle rn = 2 






37-25 
32-25 


U U ^g ^ I 


6h 




18-5 


Radius of back corners vx = 


4 




4 


Then, 122-75 x 79-5 = 9758-525 square 


inches, 


as a 


rectande. 


And 18-5^ X -7854 = 268-8 


a 


area of circle 


formed by the two ends. 









Total 10027-325 '' " from which de- 

duct the area of the coke-space, and the difference of area between 
the semicircle formed by the two back corners, and that of a rect- 
angle of equal length and breadth ; 

Then 46-75 x 37-25 = 1731-4375 area of r, w. 



32-25^ X -7854 



408-4 area of half the circle rn. 



Radius of back corners = 4 inches 



consequently < 

8 X 4 = 32 - 

the corners. 



25-13, the semicircle's area: 



X -7854 

25-13 = 6-87 inches taken off 



rounding 



Hence, 1731-4375 + 408-4 + 6-87 = 2146-707, and 

10027-235 - 2146-707 = 7880-618 square inches, or 

whole area in plan, 
7880-618 X 32 the depth = 252179-776 cubic inches, 
and 252179-776 divided by 231 gives 1091-6873 the 
content in gallons. 



MENSUEATION OF TIMBER. 



93 



Tables hy ivhich to facilitate the 3Iensu7'ation of Timher 
1. Flat or Board Measure. 



Breadth in 


Area of a 


Breadth in 


Area of a 


Breadth in 


Area of a 


inches. 


lineal foot. 


inches. 


lineal foot. 


inches. 


lineal foot. 


1 


•0208 


4 


•3334 


8 


•6667 


1 


•Oil7 


4i 


•3542 


8i 


•6875 


f 


•0625 


^ 


•375 


^ 


'•7084 


1 


•0884 


4f 


•3958 


8J 


•7292 


11- 


•1042 


5 


•4167 


9 


•75 


H 


•125 


5i 


•4375 


91 


•770^ 


If 


•1459 


5i 


•4583 


9-^ 


•7917 


2 


•1667 


61 


•4792 


n 


•8125 


2i 


•1875 


6 


•5 


10 


•8384 


^ 


•2084 


6i 


•5208 


101 


•8542 


2f 


•2292 


6J 


•5416 


m 


•875 


3 


•25 


6f 


•5625 


10| 


•8959 


H 


•2708 


7 


•5833 


11 


•9167 


H 


•2916 


7|- 


•6042 


iij 


•9875 


3-1 


•3125 


72" 


•625 


11^ 


•9583 






7|- 


•6458 


iif 


•9792 



Application and Use of the Table. 

Required the number of square feet in a board or plank 16J feet 
in length and 9| inches in breadth. 

Opposite 9f is -8125 x 16-5 = 13-4 square feet. 
A board 1 foot 2| inches in breadth, and 21 feet in length ; what 
is its superficial content in square feet ? 

Opposite 2f is -2292, to which add the 1 foot ; then 
1-2292 X 21 = 25-8 square feet. 

In a board 15| inches at one end, 9 inches at the other, and 
14| feet in length, how many square feet ? 
15.5 4. 9 
~— = 121 or 1-0208; and 1-0208 X 14-5 = 14-8 sq. ft. 

The solidity of round or unsquared timber may be found with 
much more accuracy by the succeeding Rule : — Multiply the square 
of one-fifth of the mean girth by twice the length, and the product 
will be the solidity, very near the truth. 

A piece of timber is 30 feet long, and the mean girth is 128 in- 
ches, what is the solidity ? 

128 
-^=25-6. 

25-62 X 60 
Then Y^ = 273-06 cubic feet. 

This is nearer the truth than if one-fourth the girth be em- 
ployed. 



94 



THE PRACTICAL MODEL CALCULATOR. 



2. Cubic or Solid Measure. 



Slean >4 


Cubic feet 


Mean ^ 


Cubic feet 


Mean )i 


Cubic feet 


Mean X 


Cubic feet 


girt in 


in each 


girt in 


in each 


gu-t in 


in each 


girt in 


in each 


inches. 


lineal foot. 


inches. 


lineal foot. 


inches. 


lineal foot. 


inches. 


lineal foot. 


6 


•25 


12 


1 


18 


2-25 


24 


4 


H 


•272 


121 


1-042 


181 


2-313 


241 


4-084 


^ 


•294 


121- 


1-085 


18* 


2-376 


24* 


4^168 


6| 


•317 


12| 


1-129 


18| 


2-442 


24f 


4-254 


7 


•340 


13 


M74 


19 


2-506 


25 


4-34 


7-. 


•364 


131 


1^219 


191 


2-574 


251 


4-428 


71 


•39 


13* 


1-265 


191 


2-64 


25J 


4-516 




•417 


13f 


1-313 


19f 


2-709 


25f 


4-605 


8* 


•444 


14 


1-361 


20 


2-777 


26 


4-694 


81 


•472 


141 


1-41 


201 


2-898 


261 


4-785 


8* 


•501 


14* 


1-46 


201 


2-917 


26i 


4-876 


8f 


•531 


14f 


1-511 


20f 


2-99 


26f 


4-969 


9^ 


•562 


15 


1-562 


21 


3-062 


27 


5-062 


n 


•594 


151 


1-615 


211 


3-136 


271 


5-158 


4 


•626 


151 


1-668 


21* 


3-209 


27* 


5-252 


9| 


•659 


15f 


1-772 


21f 


3-285 


27f 


5-348 


10 


•694 


16 


1-777 


22 


3-362 


28 


5-444 


101 


•73 


161 


1-833 


221 


3-438 


281 


5-542 


101 


•766 


16* 


1-89 


22J 


3-516 


28-1 


5-64 


lOf 


•803 


16| 


1-948 


22f 


3-598 


28| 


5-74 


11 


•84 


17 


2-006 


23 


3-673 


29 


5-84 


Hi 


•878 


17i 


2-066 


231 


3-754 


291 


5-941 


11* 


•918 


17* 


2-126 


23J 


3-835 


29J 


6-044 


llf 


•959 


17| 


2-187 


23f 


3-917 


29| 


6-146 



In the cubic estimation of timber, custom has established the 
rule of J, the mean girt being the side of the square considered as 
the cross sectional dimensions ; hence, multiply the number of cubic 
feet by lineal foot as in the Table of Cubic Measure opposite the 
+ girt, and the product is the solidity of the given dimensions in 
cubic feet. 

Suppose the mean J girt of a tree 21^ inches, and its length 
16 feet, what are its contents in cubic feet ? 

3-136 X 16 = 50-176 cubic feet. 

Battens, Deals, and Planks are each similar in their various 
lengths, but differing in their widths and thicknesses, and hence 
their principal distinction : thus, a batten is 7 inches by 2 J, a deal 
9 by 3, and a plank 11 by 3, these being what are termed the 
standard dimensions, by which they are bought and sold, the length 
of each being taken at 12 feet ; therefore, in estimating for the 
proper value of any quantity, nothing more is required than their 
lineal dimensions, by which to ascertain the number of times 12 feet, 
there are in the given whole. 

Suppose I wish to purchase the following : 

7 of 6 feet 6 X 7 = 42 feet 
5 14 14 X 5 = 70 

11 19 19 X 11 = 209 

and 6 21 21 X 6 = 126 

12 ) 447 ) 37-25 standard deals. 



MENSURATION OF TIMBER. 



95 



Table showing the number of Lineal Feet of Scantling of various 
dimensions, which are equal to a Cubic Foot. 





Inches. 




Ft. In. 




Indies. 




Ft. 


In. 


t>» 


Inches. 




Ft. In. 


2 


36 


4 


9 





n 


2 




2i 




28 9 




4} 




8 





-^ 


10 




2 5 




3 




24 




5' 




7 


2 


© 


lOJ 




2 3 




H 




20 7 




5J 




6 


6 


f4 




11 




2 2 




4 




18 




6 




6 





ti 


llj 




2 1 




4i 




16 


>^ 


6J 




6 


6 


CO 


12 




2 




5 




14 5 
13 1 


-^ 






5 
4 


1 

9 










^ 




g 


7 
7-5- 






7 




2 11 




6 




12 


1 


8" 




4 


6 




7i- 




2 9 




6J 




11 1 


.r-, 


8J 




4 


3 




8 




2 6 





7 




10 3 


■^ 


9 




4 





^ 


8* 




2 5 




7i 




9 7 




9j 




3 


9 


M 


9 




2 3 




8 




9 




10^ 




3 


7 


© 


9^ 




2 2 




^ 




8 6 




lOJ 




3 


5 


1 


10 




2 1 




9 




8 




11 




3 


3 


!>. 


10| 




1 11 




9J 




7 7 




Hi 




3 


2 




11 




1 10 




10 


:S 


7 3 




12 


^ 


3 







llj 


a 


1 9 




10, 


fcfl 


6 10 

6 6 






fcC 








12 


bC 


1 8 




a 




5 




5 


9 




© 








12 


.s 


6 4 
6 




? 


© 


5 
4 


3 
10 




8 


© 


2 3 
2 1 












? 




4 
4 


5 
1 




9 

9J 


^ 

■& 


2 
1 10 




3 


16 




3^ 


o 
h 


13 8 


>> 


7i 


© 


3 


10 


© 


10 


© 


1 9 




4 




12 


ri2 


8^ 




3 


7 


J 


lOJ 




1 8 




t^ 




10 8 


© 


f 




3 


5 


00 


11 




1 7 




5 




9 7 


p5 




3 


2 




Hi 




1 7 




^* 




9 
8 


o 


1^^ 




3 

2 



10 




12 




1 6 








>^ 


6J 




7 4 




10* 




2 


9 




9 




1 9 


^ 


7 




6 10 




J.V2 
11 




2 


8 


^ 


n 




1 8 


■-1 


7^ 




6 4 




12 




2 


6 


CQ 


10 




1 7 


f5 


8 




6 






2 


4 


.j:^ 


lOJ 




1 6 


CO 


8-J 
9 




5 8 
5 4 












OS 


11 

Hi 




1 5 
1 4 




6 


4 







9^ 




5 


^ 


6J 




3 


8 




12 




1 4 




10 
lOJ 




4 10 
4 6 


7 
7i 




3 
3 


5 
2 


















^ 


10 




1 5 




11 




4 4 




8 




3 







10* 




1 4 




lli- 




4 2 


o 


8J 




2 


10 


.9 


ir 




1 4 




12 




4 




9 







« 





lU 




1 3 



Hewn and sawed timber are measured by the cubic foot. The 
unit of board measure is a superficial foot one inch thick. 

To measure round timber. — Multiply the length in feet by the 
square of \ of the mean girth in inches, and the product divided 
by 144 gives the content in cubic feet. 

The \ girths of a piece of timber, taken at five points, equally 
distant from each other, are 24, 28, 33, 35, and 40 inches; the 
length 30 feet, what is the content ? 

24 + 28 + 33 + 35 + 40 _ 



Then 



32^ 



5 

X 30 
144 



96 



THE PRACTICAL MODEL CALCULATOE. 



Table containing the Super jicies and Solid Content of Spheres^ from 
1 to 12, and advancing hy a tenth. 



Diam. 


Superficies. 


Solidity. Dia 


m. 


Superfi(Jies. 


Solidity. 


Diam. 


Superficies. 


Solidity. 


1-0 


3-1416 


•5236 4 


7 


69-3979 


54-3617 


8-4 


221-6712 


310-3398 


•1 


3-8013 


•6969 


8 


72-3824 


57-9059 


•5 


226-9806 


321-5558 


•2 


4-5239 


•9047 


9 


75-4298 


61-6010 


•6 


232-3527 


333-0389 


•3 


5-3093 


1-1503 5 





78-5400 


65-4500 


•7 


237-7877 


344-7921 


•4 


6-1575 


1-4367 


1 


81-7130 


69-4560 


•8 


243-2855 


356-8187 


•5 


7-0686 


1-7671 


2 


84-9488 


73-6223 


•9 


248-8461 


369-1217 


•6 


8-0424 


2-1446 


3 


88-2475 


77-9519 


9-0 


254-4696 


381-7044 


•7 


9-0792 


2-5724 


4 


91-6090 


82-4481 


•1 


260-1558 


394-5697 


•8 


10-1787 


3-0536 


5 


95-0334 


87-1139 


•2 


265-9130 


407-7210 


•9 


11-3411 


3-5913 


6 


98-5205 


91-9525 


•3 


271-7169 


421-1613 


2-0 


12-5664 


4-1888 


7 


102-0705 


96-9670 


•4 


277-5917 


434-8937 


•1 


13-8544 


4-8490 


8 


105-6834 


102-1606 


•5 


283-5294 


448-9215 


•2 


15-2053 


5-5752 


9 


109-3590 


107-5364 


•6 


289-5298 


463-2477 


•3 


16-6190 


6 3706 6 





113-0976 


113-0976 


•7 


295-5931 


477-7755 


•4 


18-0956 


7-2382 , 


1 


116-8989 


118-8472 


•8 


301-7192 


492-8081 


•5 


19-6350 


8-1812 


2 


120-7631 


124-7885 


•9 


307-9082 


508-0485 


•6 


21-2372 


9-2027 


3 


124-6901 


130-9246 


10-0 


314-1600 


523-6000 


•7 


22-9022 


10-3060 


4 


128-6799 


137-2585 


•1 


320-4746 


539-4656 


•8 


24-6300 


11-4940 


5 


132-7326 


143-7936 


•2 


326-8520 


555-6485 


•9 


26-4208 


12-7700 


6 


136-8480 


150-5329 


•3 


333-2923 


572-1518 


3.0 


28-2744 


14-1372 


7 


141-0264 


157-4795 


•4 


339-7954 


588-9784 


•1 


30-1907 


15-5985 


8 


145-2675 


164-6365 


•5 


346-3614 


606-1324 


•2 


32-1699 


17-1573 


9 


149-5715 


172-0073 


•6 


352-9901 


623-6159 


•3 


34-2120 


188166 7 





153-9384 


179-5948 


•7 


359-6817 


641-4325 


•4 


36-3168 


20-5795 


1 


158-3680 


187-4021 


•8 


366-4362 


659-5852 


•5 


38-4846 


22-4493 


2 


162-8605 


195-4326 


•9 


373-2534 


678-0771 


•6 


40-7151 


24-4290 


3 


167-4158 


203-6893 


11-0 


380-1336 


696-9116 


•7 


43-0085 


26-5219 


4 


172-0340 


212-1752 


•1 


387-0765 


716-0915 


•8 


45-3647 


28-7309 


5 


176-7150 


220-8937 


•2 


394-0823 


735-6200 


•9 


47-7837 


31-0594 


6 


181-4588 


229-8478 


•3 


401-1509 


755-5008 


4-0 


60-2656 


33-5104 


7 


186-2654 


239-0511 


•4 


408-2823 


775-7364 


•1 


52-8102 


36-0870 


8 


191-1349 


248-4754 


•5 


415-4766 


796-3301 


•2 


55-4178 


38-7924 


9 


196-0672 


258-1552 


•6 


422-7336 


817-2851 


•3 


58-0881 


41-6298 8 





201-0624 


268-0832 


•7 


430-0536 


838-6045 


•4 


60-8213 


44-6023 


1 


206-1203 


278-2625 


•8 


437-4363 


860-2915 


•5 


63-6174 


47-7130 


2 


211-2411 


288-6962 


•9 


444-8819 


882-3492 


•6 


66-4782 


50-9651 


3 


216-4248 


299-3876 


12-0 


452-3904 


904-7808 







To reduce Solid Inches into Solid Feet. 








1728 Solid Inches to 


one Solid Foot. 






Feet. Indies. 


Feet. 


Inches. 


Feet. 


Inches. 


Feet. 


Inches. 


Feet. 


Inches. 


Feet. 


Inches. 


1 = 1728 


18= 


=31104 


35 = 


=60480 


52 = 


=88956 


69 = 


=119232 


85 = 


=146880 


2 3456 


19 


32832 


36 


62208 


53 


91584 


70 


120960 


86 


148608 


3 5184 


20 


34560 


37 


63936 


54 


93312 


71 


122688 


87 


150336 


4 6912 


21 


36288 


38 


65664 


55 


95040 


72 


124416 


88 


152064 


5 8640 


22 


38016 


39 


67392 


56 


96768 


73 


126144 


89 


163792 


6 10368 


23 


39744 


40 


69120 


57 


98496 


74 


127872 


90 


155620 


7 12096 


24 


41472 


41 


70848 


58 


100224 


75 


129600 


91 


167248 


8 13824 


25 


43200 


42 


72576 


69 


101952 


76 


131328 


92 


158976 


9 15552 


26 


44928 


43 


74304 


60 


103680 


77 


133056 


93 


160704 


10 17280 


27 


40656 


44 


76032 


61 


105408 


78 


134784 


94 


162432 


11 19008 


28 


48384 


45 


77760 


62 


107136 


79 


136512 


95 


164160 


12 20736 


29 


50112 


46 


79488 


63 


108864 


80 


138240 


96 


165888 


13 22464 


30 


51840 


47 


81216 


64 


110592 


81 


139968 


97 


167616 


14 24192 


31 


53568 


48 


82944 


65 


112320 


82 


141696 


98 


169344 


15 25920 


32 


55296 


49 


84672 


66 


114048 


83 


143424 


99 


171072 


16 27648 


33 


67024 


50 


86400 


67 


116776 


84 


145152 


100 


172800 


17 29376 


34 


58762 


51 


88128 


68 


117504 











CUTTINGS AND EMBANKMENTS. 



97 



CUTTINGS AND EMBANKMENTS. 

The angle of repose upon railways, or that incline on which a 
carriage would rest in whatever situation it was placed, is said to 
be at 1 in 280, or nearly 19 feet per mile ; at any greater rise 
than this, the force of gravity overcomes the horizontal traction, 
and carriages will not rest, or remain quiescent upon the line, but 
will of themselves run down the line with accelerated velocity. 
The angle of practical effect is variously stated, ranging from 1 in 
75 to 1 in 330. 

The width of land required for a railway must vary with the 
depth of the cuttings and length of embankments, together with 
the slopes necessary to be given to suit the various materials of 
which the cuttings are composed : thus, rock will generally stand 
when the sides are vertical ; chalk varies from |- to 1, to 1 to 1 ; 
gravel 1|- to 1; coal 1|- to 1 ; clay 1 to 1, &c. ; but where land 
can be obtained at a reasonable rate, it is always well to be on the 
safe side. 

The following Table is calculated for the purpose of ascertain- 
ing the extent of any cutting in cubic yards, for 1 chain, 22 yards, 
or 6Q feet in length, the slopes or angles of the sides being those 
which are most in general practice, and formation level equal 30 feet. 











Slopes 


Itol. 










Depth 
of 

cut- 
ting in 

feet. 


Half 

width 

at 
top in 

feet. 


Content 

in cubic 

yards per 

chain. 


Content 
of 1 per- 
pendicu- 
lar ft. in 
breadth. 


Content 
of 3 per- 
pendicu- 
lar ft. in 
breadth. 


Content 
of 6 per- 
pendicu- 
lar ft. in 
breadth. 


Depth 

cut- 
ting in 
feet. 


Half 

width 

at 
top in 
feet. 


Content 

in cubic 

yards per 

chain. 


Content 
of 1 per- 
pendicu- 
lar ft. in 
breadth. 


Content 
of 3 per- 
pendicu- 
lar ft. in 
breadth. 


Content 
of G per- 
pendicu- 
lar ft. in 
breadth. 


1 


16 


75-78 


2-44 


7-33 


14-67 


26 


41 


3599-11 


63-55 


190-67 


381-33 


2 


17 


156-42 


4-89 


14-67 


29-33 


27 


42 


3762-00 


65-99 


198-00 


396-00 


3 


18 


242-00 


7-33 


22-00 


44-00 


28 


43 


3969-78 


68-43 


205-33 


410-67 


4 


19 


332-44 


9-78 


29-33 


68-67 


29 


44 


4182-44 


70-88 


212-67 


425-33 


5 


20 


427-78 


12-22 


36-67 


73-33 


30 


45 


4400-00 


73-32 


220-00 


440-00 


6 


21 


528-00 


14-67 


44-00 


88-00 


31 


46 


4622-44 


75-77 


227-33 


454-67 


7 


22 


633-11 


17-11 


51-33 


102-67 


32 


47 


4849-78 


78-22 


234-67 


469-33 


8 


23 


743-11 


19-56 


58-67 


117-33 


33 


48 


5082-00 


80-67 


242-00 


484-00 


9 


24 


858-00 


22-00 


66-00 


132-00 


34 


49 


5319-11 


83-11 


249-33 


498-67 


10 


25 


977-78 


24-44 


73-33 


146-67 


35 


50 


5561-11 


85-55 


256-67 


513-33 


11 


26 


1102-44 


26-89 


80-67 


161-33 


36 


51 


5808-00 


88-00 


264-00 


528-00 


12 


27 


1232-00 


29-33 


88-00 


176-00 


37 


52 


6059-78 


90-44 


271-33 


542-67 


13 


28 


1366-44 


31-78 


95-33 


190-67 


38 


53 


6316-44 


92-39 


278-67 


557-33 


14 


29 


1505-78 


34-22 


102-67 


205-33 


39 


54 


6578-00 


95-33 


286-00 


572-00 


15 


30 


1650-00 


36-66 


110-00 


220-00 


40 


55 


6844-44 


97-77 


293-33 


586-67 


16 


31 


1799-11 


39-11 


117-33 


234-67 


41 


56 


7115-78 


100-22 


300-67 


601-33 


17 


32 


1953-11 


41-55 


124-67 


249-33 


42 


57 


7392-00 


102-66 


308-00 


616-00 


18 


33 


2112-00 


43-99 


132-00 


264-00 


43 


58 


7673-11 


105-11 


315-33 


630-67 


19 


34 


2275-78 


46-44 


139-33 


278-67 


44 


59 


7959-11 


107-55 


322-67 


645-33 


20 


35 


2444-44 


48-89 


146-67 


293-33 


45 


60 


8250-00 


109-99 


330-00 660-00 


21 


36 


2618-00 


51-33 


154-00 


308-00 


46 


61 


8545-78 


112-44 


337-33674-67 


22 


37 


2796-44 


53-77 


161-33 


322-67| 


47 


62 


8846-44 


114-88 


344-67 689-33 


23 


38 


2979-78 


56-21 


168-67 


337-33 


48 


63 


9152-00 


117-33 


352-00 704-00 


24 


39 


3168-00 


58-66 


176-00 


352-00 


49 


64 


9462-44 


119-77 


359-33 718-67 


25 


40 


3361-11 


61-10 


183-33 


366-67 


50 


65 


9777-78 


122-21 


366-671733-33 






I 








7 













98 



THE PRACTICAL MODEL CALCULATOR. 



Slopes 1^ to 1. 



jDeutii 


Half 




1 <"f" 


width 


Content 


cut- 


at 


in cubic 


'tingia 


top in 


yards per 


1 feet. 


teet. 


chain. 


1 


16i 


77-00 


2 


18 


161-33 


3 


19,^ 


253-00 


4 


21 


352-00 


5 


22* 


453-33 


6 


24 


572-00 


7 


251 


693-00 


8 


27 


821-33 


9 


28* 


957-00 


10 


30 


1100-00 


11 


31i 


1250-33 


12 


33 


1408-00 


13 


341 


1573-00 


14 


36 


1745-33 


15 


37* 


1925-00 


16 


39 


2112-00 


17 


40* 


2306-33 


18 


42 


2508-00 


19 


43* 


2717-00 


20 


45 


2933-33 


21 


46* 


3157-00 


22 


48 


3388-00 


23 


49* 


3626-33 


24 


51 |3872-00 


25 


52*!4125-00 



Content 
of 1 per- 
pendicu- 
lar ft. in 
breadth. 



2-44 
4-89 
7-33 
9-78 
12-22 
14-67 
17-11 
19-56 
22-00 
24-44 
26-89 
29-33 
31-78 
34-22 
36-66 
39-11 
41-55 
43-99 
46-44 
48-89 
51-33 
53-77 
56-21 
58-66 
61-10 



Content 
of 3 per- 
pendicu- 
lar ft. in 
breadth. 



Content 
of 6 per- 
pendicu- 
lar ft. in 
breadth. 



f66 14 

14-67 29 

22-00 44 

29-33! 58' 

36-67 73' 

44-00 88' 

51-33 102' 

58-67;il7' 

66-00 132' 

73-38|l46' 

80-67 161- 

88-00176' 

95-33il90' 

102-67'205' 

110-00:220' 

117-33'234' 

124-67|249- 

132-00|264- 

139-33278- 



■67i 
33 

oo: 

67i 
33 

oo; 

67l 
33| 
00' 
67i 
33 
00 
671 
33' 

oo: 

67i 
33 
00 
67t 
146-671293-33^ 



154-00308 
161-33,322 
168-67'337 
176-00852 
183-331366 



Depth 
of 

cut- 
ting in 

feet. 



top in 



54 

551 

57 

58* 

60" 

611 

63 

64* 

66" 

671 

69 

70J 

72 

73* 

lb 



791 
81 
821 
84 



90 



Content 

in cubic 

yards per 

chain. 



4385' 
4653' 
4928' 
6210 
5500' 
5797' 
6101' 
6413' 
6732' 
7058' 
7392' 
7733' 
8081' 
8437' 
8800' 
9170' 
9548' 



Content 
of 1 per- 
pendicu- 
lar ft. in 
breadth. 



Content 
of 3 per- 
pendicu- 
lar ft. in 
breadth. 



63-55 
65-99 
68-43 
70-88! 
73-32 
75-77 
78-22i 
80-67| 
83-11' 
85-55 
88-00 
90-44' 
92-39! 
95-33! 
97-771 
100-221 
102-66 



Content 
of 6 per- 
pendicu- 
lar ft. in 
breadth. 



190-67 381-38 
198-00 396-00 



53-001105-11 



10325-33 
10725-00 
11132-00 
11546-88 
11968-00 
12397-00 
12838-33 



107-55 
109-99 
112-44 
114-88' 
117-33 
119-77; 
122-21 



205-33 
212-67 
220-00 
227-33 
234-67 
242-00 
249-38 
256-67 
264-00 
271-33 
278-67 
286-00 
293-33 
300-67 
308-00 
315-33 
322-67 



410-67 
425-33 
440-00 
454-67 
469-33 
484-00 
498-67 
513-83 
528-00 
542-67 
557-33 
572-00 
586-67 
601-88 
616-00 
630-67 
645-33 



330-00!660-00 
337-38, 674-67 
844-67 689-83 



352-00 
359-38 
366-67 



704-00 
718-67 
738-33 













Slope 


s 2 1^0 1. 










Depth 


Half 1 Content 


Content 


Content 


Depth 


Half 




Content 


Content 


Content 


of 


width 1 Content of 1 per- 


of 3 per- 


of 6 per- 


of 


width 


Content 


of 1 per- 


of 3 per- 


of 6 per- 


cut- 


at j in cubic Ipendicu- 


pendicu- 


pendicu- 


cut- 


at 


in cubic 


pendicu- 


pendicu- 


pendicu- 


ting in 


top in 


yards per :lar ft. in 


lar ft. in 


lar ft. in 


ting in 


top in 


yards per 


lar ft. in 


lar ft. in 


lar ft. in 


feet. 
1 


feet. 


chain, jbreadth. 


breadth. 


breadth. 


feet. 


feet. 


chain. 


breadth. 


breadth. 


breadth. 


17 


78-22 


2-44 


7-38 


14-67 


26 


67 


6211-65 


63-55 


190-67!381-33 


2 


19 


166-22 


4-89 


14-67 


29-38 


27 


69 


5544-00 


65-99 


198-00396-00 


3 


21. 


264-00 


7-38 


22-00 


44-00 


28 


71 


6886-22 


68-43 


205-38410-67 


4 


23 1 371-55 


9-78 


29-83 


68-67 


29 


73 


6238-22 


70-88 


212-67425-33 


5 


25 488-89 


12-22 


86-67 


73-38 


30 


75 


6600-00 


73-32 


220-00 440-00 


6 


27 616-00 


14-67 


44-00 


88-00 


31 


77 


6971-55 


75-77 


227-33 454-67 


7 


29 ! 752-89 


17-11 


51-83 


102-67 


82 


79 


7852-89 


78-22 


234-67 469-83 


8 


31 899-55 


19-56 


58-67 


117-33 


33 


81 


7744-00 


80-67 


242-00484-00 


9 


38 1056-00 


22-00 


66-00 


132-00 


34 


83 


8144-89 


83-11 


249-38 498-67 


10 


35 1222-22 


24-44 


73-33 


146-67 


35 


85 


8555-55 


85-55 


256-67!518-33 


11 


87 I1398-22 


26-89 


80-67 


161-83 


36 


87 


8976-00 


88-00 


264-00528-00 


12 


39 |1584-00 


29-83 


88-00 


176-00 


87 


89 


9406-22 


90-44 


271-33.542-67 


13 


41 1779-55 


31-78 


95-83 


190 67 


38 


91 


9846-22 


92-39 


278-67 557-33 


14 


43 11984-89 


34-22 


102-67 


205-33 


39 


93 


10296-00 


95-33 


286-00572-00 


15 


45 i2200-00 


36-66 


110-00 


220-00 


40 


95 


10755-55 


97-77 


293-83:586-67 


16 


47 2424-89 


89-11 


117-83 


234-67 


41 


97 


11224-89 100-22 


300-67601-33 


17 


49 2659-55 


41-55 


124-67i249-33 


42 


99 


11704-00il02-66 


308-00616-00 


18 


51 i2904-00 


43-99 


132-00264-00 


43 


101 


12192-891105-11 


315-33 630-67 


19 


53 3158-22 


46-44 


189-83278-67 


44 


103 


12691-55;i07-65 


322-67 645-33 


20 


55 84-2222 


48-89 


146-67293-33 


45 


105 


13200-00;i09-99 


880-00 660-00 


21 


57 :3696-00 


51-33 


154-00 308-00! 


46 


107 


18718-22 112-44 


837-33!674-67 


22 


59 '3979-55 


53-77 


161-33'322-67i 


47 


109 


14246-22 114-88 


344-67 689-33 


23 


61 4272-89 


56-21 


168-67|337-33i 


48 


111 


14784-00 117-33 


352-00 704-00 


24 


63 4576-00 


58-66 


176-00 


852-00 


49 


113 


15331-55 119-77 


359-33 718-67 


25 


65 14888-89 


61-10 183-33 


366-67 


50 


115 


15888-89122-21 

1 


366-67 733-33 

1 



CUTTINGS AND EMBANKMENTS. 99 

Bj the fourth, fifth, and sixth columns in each table, the number 
of cubic yards is easily ascertained at any other width of formation 
level above or below 30 feet, having the same slopes as by the 
tables, thus : — 

Suppose an excavation of 40 feet in depth, and 33 feet in width 
at formation level, whose slopes or sides are at an angle of 2 to 1, 
required the extent of excavation in cubic yards : 

10755-55 + 293-33 = 11048-88 cubic yards. 

The number of cubic yards in any other excavation may be as- 
certained by the following simple rule : 

To the width at formation level in feet, add the horizontal length 
of the side of the triangle formed by the slope, multiply the sum 
by the depth of the cutting, or excavation, and by the length, also 
in feet ; divide the product by 27, and the quotient is the content 
in cubic yards. 

Suppose a cutting of any length, and of which take 1 chain, its 
depth being 14|^ feet, width at the bottom 28 feet, and whose sides 
have a slope of 1|- to 1, required the content in cubic yards : 

14-5 X 1-25 = 18-125 + 28 x 14 = 645-75 x 6Q = 
42619-5 
— 07 = 1578*5 cubic yards. 

I {(5 + rh') h' + {b + rh) A + 4 [5 + /^^]^-^} 

gives the content of any cutting. In words, this formula will be : — 
To the area of each end, add four times the middle area ; the sum 
multiplied by the length and divided by 6 gives the content. The 
breadth at the bottom of cutting = h ; the perpendicular depth of 
cutting at the higher end = h ; the perpendicular depths of cutting 
at the lower end = 7i' ; ?, the length of the solid ; and rh' the ratio 
of the perpendicular height of the slope to the horizontal base, mul- 
tiplied by the height h'. rh, the ratio r, of the perpendicular 
height of the slope, to the horizontal base, multiplied by the 
height h. 

Let 6 = 30 ; 7i = 50 ; A' = 20 ; Z = 84 feet ; and 2 to 5 or | 
the ratio of the perpendicular height of the slope to the horizontal 
base : 

84 r 50 4- 20 

-^ I (30 + f X 20) 20 + (30 + f X 50) 50 + 4 [30 + f ^^] 

5^jf^| = 14 I 38 X 20 + 50 X 50 -f 4 X 44 X 35 I = 131880 

cubic feet. — ^7 — ~ 4884-44 cubic yards. 

This rule is one of the most useful in the mensuration of solids, 
it will give the content of any irregular solid very nearly, whether 
it be bounded by right lines or not. 



100 



THE PRACTICAL MODEL CALCULATOR. 



Table of Squares, Cubes, Square and Cube Boots of Numbers. 



Ntunber. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Eeciprocals. 


1 


1 


1 


1-0000000 


1-0000000 


•100000000 


2 


4 


8 


1-4142136 


1-2599210 


•500000000 


3 


9 


27 


1-7320508 


1-4422496 


•333333333 


4 


16 


64 


2-0000000 


1-5874011 


•250000000 


5 


25 


125 


2-2360680 


1-7099759 


■200000000 


6 


36 


216 


2-4494897 


1-8171206 


•166666667 


7 


49 


343 


2-6457513 


1-9129312 


•142857143 


8 


64 


512 


2-8284271 


2-0000000 


•125000000 


9 


81 


729 


3-0000000 


2-0800837 


•111111111 


10 


100 


1000 


3-1622777 


2-1544347 


•100000000 


11 


121 


1331 


3-3166248 


2-2239801 


•090909091 


12 


144 


1728 


3-4641016 


2-2894286 


•083333333 


13 


169 


2197 


3-6055513 


2-3513347 


•076923077 


14 


196 


2744 


3-7416574 


2-4101422 


•071428571 


15 


225 


3375 


3-8729833 


2-4662121 


•066666667 


16 


256 


4096 


4-0000000 


2-5198421 


•062500000 


17 


289 


4913 


4-1231056 


2-5712816 


•058823529 


18 


324 


6832 


4-2426407 


2-6207414 


•055555556 


19 


361 


6859 


4-3588989 


2-6684016 


•052631579 


20 


400 


8000 


4-4721360 


2-7144177 


•050000000 


21 


441 


9261 


4-5825757 


2-7589243 


•047619048 


22 


484 


10648 


4-6904158 


2-8020393 


•045454545 


23 


529 


12167 


4-7958315 


2-8438670 


•043478261 


24 


576 


13824 


4-8989795 


2-8844991 


•041666667 


25 


625 


15625 


5-0000000 


2-9240177 


•040000000 


26 


676 


17576 


5-0990195 


2-9624960 


•038461538 


27 


729 


19683 


5-1961524 


3-0000000 


•037037037 


28 


784 


21952 


5-2915026 


3-0365889 


•035714286 


29 


841 


24389 


5-3851648 


3-0723168 


•034482759 


30 


900 


27000 


5-4772256 


3-1072325 


•033333333 


31 


961 


29791 


5-5677644 


3-1413806 


•032258065 


32 


1024 


32768 


5-6568542 


3-1748021 


•031250000 


33 


1089 


35937 


5-7445626 


3-2075343 


•030303030 


34 


1156 


39304 


5-8309519 


3-2396118 


•029411765 


35 


1225 


42875 


5-9160798 


3-2710663 


•028571429 


36 


1296 


46656 


6-0000000 


3-3019272 


•027777778 


37 


1369 


60653 


6-0827625 


3-3322218 


•027027027 


38 


1444 


54872 


6-1644140 


3-3619754 


•026315789 


39 


1521 


69319 


6-2449980 


3-3912114 


•025641026 


40 


1600 


64000 


6-3245553 


3-4199519 


■025000000 


41 


1681 


68921 


6-4031242 


3-4482172 


•024390244 


42 


1764 


74088 


6-4807407 


3-4760266 


•023809524 


43 


1849 


79507 


6-5574385 


3-5033981 


•023255814 


44 


1936 


85184 


6-6332496 


3-5303483 


•022727273 


45 


2025 


91125 


6-7082039 


3-5568933 


•022222222 


46 


2116 


97336 


6-7823300 


3-5830479 


•021739130 


47 


2209 


103823 


6-8556546 


3-6088261 


•021276600 


48 


2304 


110592 


6-9282032 


3-6342411 


•020833333 


49 


2401 


117649 


7-0000000 


3-6593057 


•020408163 


60 


2500 


125000 


7-0710678 


3-6840314 


•020000000 


51 


2601 


132651 


7-1414284 


3-7084298 


•019607843 


52 


2704 


140608 


7-2111026 


3-7325111 


•019230769 


53 


2809 


148877 


7-2801099 


3-7562858 


•018867925 


54 


2916 


157464 


7-3484692 


3-7797631 


•018518519 


55 


3025 


166375 


7-4161985 


3-8029525 


•018181818 


56 


3136 


175616 


7-4833148 


3-8258624 


•017857143 


57 


3249 


185193 


7-5498344 


3-8485011 


•017543860 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



101 



Number. | 


Squares. | 


Cubes. 


, Square Koots. 


Cube Roots. 


Reciprocals. 


58 


3364 


195112 


7-6157731 


3-8708766 


017241379 


59 


3481 


205379 


7-6811457 


3-8929965 


016949153 


60 


3600 


216000 


7-7459667 


3-9148676 


016666667 


61 


3721 


226981 


7-8102497 


3-9304972 


016393443 


62 


3844 


238328 


7-8740079 


3-9578915 


016129032 


63 


3969 


250047 


7-9372539 


3-9790571 


015873016 


64 


4096 


202144 


8-0000000 


4-0000000 


015625000 


65 


4225 


274625 


8-0622577 


4-0207256 


015384015 


66 


4356 


287496 


8-1240384 


4-0412401 


015151515 


67 


4489 


300763 


8-1853528 


4-0615480 


014925373 


68 


4624 


314432 


8-2462113 


4-0816551 


014705882 


69 


4761 


328509 


8-3066239 


4-1015661 


014492754 


70 


4900 


343000 


8-3666003 


4-1212853 


014285714 


71 


5041 


357911 


8-4261498 


4-1408178 


014084517 


72 


5184 


373248 


8-4852814 


4-1601676 


013888889 


73 


5329 


389017 


8-5440037 


4-1793390 


013698630 


74 


5476 


405224 


8-6023253 


4-1983364 


013513514 


75 


5625 


421875 


8-6602540 


4-2171633 


013333333 


76 


5776 


438976 


8-7177979 


4-2358236 


013157895 


77 


5929 


456533 


8-7749644 


4-2543210 


012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


012820513 


79 


6241 


493039 


8-8881944 


4-2908404 


012658228 


80 


6400 


512000 


8-9442719 


4-3088695 


012500000 


81 


6561 


531441 


9-0000000 


4-3267487 


012345679 


82 


6724 


651368 


9-0553851 


4-3444815 


012195122 


83 


6889 


571787 


9-1104336 


4-3620707 


012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


011904762 


85 


7225 


614125 


9-2195445 


4-3968296 


011764706 


86 


7396 


630056 


9-2736185 


4-4140049 


011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


011494253 


88 


7744 


681472 


9-3808315 


4-4470692 


011363036 


89 


7921 


704969 


9-4339811 


4-4647451 


011235955 


90 


8100 


729000 


9-4868330 


4-4814047 


011111111 


91 


8281 


753571 


9-5393920 


4-4979414 


010989011 


92 


8464 


778688 


9-5916630 


4-5143574 


010869565 


93 


8649 


804357 


9-6436508 


4-5306549 


010752088 


94 


8836 


830584 


9-6953597 


4-5468359 


010638298 


95 


9025 


857374 


9-7467943 


4-5629026 


010526316 


96 


9216 


884736 


9-7979590 


4-5788570 


010416667 


97 


9409 


912673 


9-8488578 


4-5947009 


010309278 


98 


9604 


941192 


9-8994949 


4-6104303 


010204082 


99 


9801 


970299 


9-9498744 


4-6260650 


010101010 


100 


10000 


1000000 


10-0000000 


4-6415888 


010000000 


101 


10201 


1030301 


10-0498756 


4-6570095 


009900990 


102 


10404 


1061208 


10-0995049 


4-6723287 


009803922 


103 


10609 


1092727 


10-1488916 


4-6875482 


009708738 


104 


10816 


1124864 


10-1980390 


4-7026694 


009615385 


105 


11025 


1157625 


10-2469508 


4-7176940 


009523810 


106 


11230 


1191016 


10-2956301 


4-7326235 


009433902 


107 


11449 


1225043 


10-3440804 


4-7474594 


009345794 


108 


11664 


1259712 


10-3923048 


4-7622032 


009259259 


109 


11881 


1295029 


10-4403065 


4-7768562 


009174312 


110 


12100 


1331000 


10-4880885 


4-7914199 


009090909 


111 


12321 


1367631 


10-5356538 


4-8058995 


009009009 


112 


12544 


1404928 


10-5830052 


4-8202845 


008928571 


113 


12769 


1442897 


10-6301458 


4-8345881 


008849558 


114 


12996 


1481544 


10-6770783 


4-8488076 


008771930 


115 


13225 


1520875 


10-7238053 


4-8629442 


008095652 


116 


13456 


1560896 


10-7703296 


4-8769990 


008020690 


117 


13689 


1601013 


10-8166538 


4-8909732 


008547009 


118 


13924 


1643032 


10-8627805 


4-9048681 


008474576 


119 


14161 


1685159 


10-9087121 


4-9186847 


008403361 



i2 



102 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


S'luare Roots. 


Cvibe Koots. 


Eeeiprocals. 


120 


14400 


1728000 


10-9544512 


4-9324242 


•008883333 


121 


14641 


1771561 


11-0000000 


4-9460874 


■008264463 


122 


14834 


1815848 


11-0453610 


4-9596757 


•008196721 


123 


15129 


1860867 


11-0905365 


4-9731898 


•008130081 


124 


15376 


1906624 


11-1355287 


4-9866310 


•008064516 


125 


15625 


1953125 


11-1803899 


5-0000000 


•008000000 


126 


15876 


2000376 


11-2249722 


5-0132979 


•007986508 


127 


16129 


2048383 


11-2694277 


5-0265257 


•007874016 


128 


16384 


2097152 


11-3137085 


5-0896842 


•007812500 


129 


16641 


2146689 


11-3578167 


5-0527743 


•007751938 


180 


16900 


2197000 


11-4017543 


5-0657970 


•007692308 


181 


17161 


2248091 


11-4455231 


6-0787531 


•007633588 


132 


17424 


2299968 


11-4891253 


5-0916434 


-007575758 


138 


17689 


2352637 


11-5825626 


6-1044687 


•007518797 


134 


17956 


2406104 


11-5758869 


5-1172299 


•007462687 


135 


18225 


2460375 


11-6189500 


5-1299278 


•007407407 


136 


18496 


2515456 


11-6619038 


5-1425632 


•007352941 


137 


18769 


2571358 


11-7046999 


5-1551867 


•007299270 


138 


19044 


2628072 


11-7473444 


5-1676493 


•007246877 


139 


19321 


2685619 


11-7898261 


6-1801015 


•007194245 


140 


19600 


2744000 


11-8321596 


5-1924941 


•007142857 


141 


19881 


2803221 


11-8743421 


5-2048279 


•007092199 


142 


20164 


2863288 


11-9163758 


5-2171084 


•007042254 


148 


20449 


2924207 


11-9582607 


6-2293215 


•006998007 


144 


20736 


2985984 


12-0000000 


5-2414828 


•006944444 


145 


21025 


3048625 


12-0415946 


5-2585879 


•006896552 


146 


21316 


3112136 


12-0830460 


5-2656374 


•006849315 


147 


21609 


3176528 


12-1243557 


5-2776321 


•006802721 


148 


21904 


8241792 


12-1655251 


5-2895725 


•006756757 


149 


22201 


3307949 


12-2065556 


5-3014592 


•006711409 


150 


22500 


3375000 


12-2474487 


5-3182928 


•006666667 


151 


22801 


3442951 


12-2882057 


6-8250740 


•006622517 


152 


23104 


3511008 


12-3288280 


5-8368033 


•006578947 


153 


23409 


8581577 


12-3693169 


5-3484812 


•006535948 


154 


23716 


3652264 


12-4096736 


5-3601084 


•006498506 


155 


24025 


3723875 


12-4498996 


5-3716854 


•006451618 


156 


24336 


8796416 


12-4899960 


5-3832126 


•006410256 


157 


24649 


3869893 


12-5299641 


6-3946907 


•006869427 


158 


24964 


3944312 


12-5698051 


6-4061202 


•006329114 


159 


25281 


4019679 


12-6095202 


5-4175015 


•006289308 


160 


25600 


4096000 


12-6491106 


5-4288352 


•006250000 


161 


25921 


4178281 


12-6885775 


6-4401218 


•006211180 


162 


26244 


4251528 


12-7279221 


6-4513618 


•006172840 


168 


26569 


4330747 


12-7671458 


5-4625556 


•006134969 


164 


26896 


4410944 


12-8062485 


5-4737087 


•006097561 


165 


27225 


4492125 


12-8452326 


5-484«066 


•006060606 


166 


27556 


4574296 


12-8840987 


6-4958647 


•006024096 


167 


27889 


4657463 


12-9228480 


5-5068784 


•005988024 


168 


28224 


4741632 


12-9614814 


5-5178484 


•005952381 


169 


28561 


4826809 


13-0000000 


5-5287748 


•005917160 


170 


28900 


4913000 


13-0884048 


5-5396583 


•005882353 


171 


29241 


5000211 


13-0766968 


6-5504991 


•005847953 


172 


29584 


5088448 


18-1148770 


5-5612978 


•005813953 


173 


29929 


5177717 


18-1529464 


5-5720546 


•005780347 


174 


30276 


5268024 


13-1909060 


6-5827702 


•005747126 


175 


30625 


5359375 


13-2287566 


6-5934447 


•005714286 


176 


80976 


5451776 


18-2664992 


6-6040787 


•005681818 


177 


31329 


6545233 


18-3041347 


5-6146724 


-005649718 


178 


31684 


5639752 


13-8416641 


5-6252268 


•005617978 


179 


32041 


5735339 


13-3790882 


5-6357408 


•005586592 


180 


82400 


5832000 


13-4164079 


5-6462162 


•005555556 


181 


32761 


5929741 


13-4536240 


5-6566528 


•005524862 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 103 



Number. 


Sauares. 


Cubes. 


Square Roots. 


Cube Koots. 


Reciprocals. 


182 


33124 


6028568 


13-4907376 


5-6670511 


•005494605 


183 


33489 


6128187 


13-5277493 


6-6774114 


•005464481 


18i 


33856 


6229504 


18-5646600 


6-6877340 


•005434783 


185 


34225 


6331625 


13-6014705 


6-6980192 


•005405405 


186 


34596 


6434856 


13-6381817 


6-7082675 


•005876344 


187 


34969 


6539203 


13-6747943 


5-7184791 


•005347594 


188 


35344 


6644672 


13-7113092 


6-7286543 


•005319149 


189 


35721 


6751269 


13-7477271 


6-7387936 


-005291005 


190 


36100 


6859000 


13-7840488- 


6-7488971 


-005263158 


191 


36481 


6967871 


13-8202750 


6-7589652 


•005235602 


192 


36864 


7077888 


13-8564065 


5-7689982 


•005208333 


193 


37249 


7189517 


13-8924400 


5-7789966 


•005181347 


194 


37636 


7301384 


13-9283883 


6-7889604 


•005154639 


195 


38025 


7414875 


13-9642400 


5-7988900 


•005128205 


196 


38416 


7529536 


14-0000000 


5-8087857 


•005102041 


197 


38809 


7645373 


14-0356688 


6-8186479 


•005076142 


198 


39204 


7762392 


14-0712473 


5-8284867 


•005050505 


199 


39601 


7880599 


14-1067360 


6-8382725 


•005025126 


200 


40000 


8000000 


14-1421356 


6-8480355 


•005000000 


201 


40401 


8120601 


14-1774469 


6-8577660 


•004976124 


202 


40804 


8242408 


14-2126704 


6-8674673 


•004950495 


203 


41209 


8365427 


14-2478068 


6-8771307 


•004926108 


204 


41616 


8489664 


14-2828569 


6-8867653 


•004901961 


205 


42025 


8615125 


14-3178211 


6-8963685 


•004878049 


206 


42436 


8741816 


14-3527001 


6-9059406 


•004854369 


207 


42849 


8869743 


14-3874946 


5-9154817 


•004830918 


208 


43264 


8998912 


14-4222051 


5-9249921 


•004807692 


209 


43681 


9129329 


14-4568323 


5-9344721 


•004784689 


210 


44100 


9261000 


14-4913767 


5-9439220 


•004761905 


211 


44521 


9393931 


14-5258390 


6-9533418 


•004739336 


212 


44944 


9528128 


14-5002198 


5-9627320 


•004716981 


213 


45369 


9663597 


14-5945195 


5-0720926 


•004094836 


214 


45796 


9800344 


14-6287388 


6-9814240 


•004672897 


215 


46225 


9938375 


14-6628783 


6-9907264 


•004651163 


216 


46656 


10077696 


14-6969385 


6-0000000 


•004629630 


217 


47089 


10218313 


14-7309199 


6-0092450 


•004608295 


218 


47524 


10360232 


14-7648231 


6-0184617 


•004587156 


219 


47961 


10503459 


14-7986486 


6-0276502 


•004566210 


220 


48400 


10648000 


14-8323970 


6-0368107 


•004545455 


221 


48841 


10793861 


14-8060687 


6-0459435 


•004524887 


222 


49284 


10941048 


14-8996644 


6-0550489 


•004504505 


223 


49729 


11089567 


14-9331845 


6-0641270 


•004484305 


224 


50176 


11239424 


14-9666295 


6-0731779 


•004464286 


225 


50625 


11390625 


15-0000000 


6-0824020 


-004444444 


226 


61076 


11513176 


15-0332964 


6-0991994 


-004424779 


227 


51529 


11697083 


15-0665192 


6-1001702 


-004405286 


228 


51984 


11852352 


15-0996689 


6-1091147 


-004385965 


229 


52441 


12008989 


15-1327460 


6-1180332 


-004366812 


230 


52900 


12167000 


15-1657509 


6-1269257 


•004347826 


231 


53361 


12326391 


15-1986842 


6-1357924 


•004329004 


232 


53824 


12487168 


15-2315462 


6-1446337 


-004310345 


233 


54289 


12649337 


15-2643375 


6-1534495 


-004291845 


234 


54756 


12812904 


15-2970585 


6-1622401 


-004273504 


235 


55225 


12977875 


15-3297097 


6-1710058 


-004255319 


236 


55696 


13144256 


15-3622915 


6-1797466 


•004237288 


237 


56169 


13312053 


15-3948043 


6-1884628 


•004219409 


238 


56644 


13481272 


15-4272486 


6-1971544 


-004201681 


239 


57121 


13651919 


15-4596248 


6-2058218 


-004184100 


240 


57600 


13824000 


15-4919334 


6-2144650 


-004166667 


241 


58081 


13997521 


15-5241747 


6-2230843 


•004149378 


242 


58564 


14172488 


15-5563492 


6-2316797 


•004132231 


243 


59049 


14348907 


15-5884573 


6-2402515 


•004115226 



104 



THE PRACTICAL MODEL CALCULATOE. 



Number. 


Squares. 


Cubes. 


Square Eoots. 


Cube Roots. 


Reciprocals. 


244 


59536 


14526784 


15-6204994 


6-2487998 


•004098361 


245 


60025 


14706125 


15-6524758 


6-2573248 


•004081633 


246 


60516 


14886936 


15-6843871 


6-2658266 


•004065041 


247 


61009 


15069223 


16-7162336 


6-2743054 


•004048583 


248 


61504 


15252992 


15-7480157 


6-2827613 


•004032258 


249 


62001 


15438249 


15-7797338 


6-2911946 


•004016064 


250 


62500 


15625000 


15-8113883 


6-2996053 


•004000000 


251 


63001 


15813251 


15-8429795 


6-3079935 


•003984064 


252 


63504 


16003008 


15-8745079 


6-3163596 


•003968254 


253 


64009 


16194277 


15-9059737 


6-3247035 


•003952569 


254 


64516 


16387064 


15-9373775 


6-3330256 


•003937008 


255 


65025 


16581375 


15-9687194 


6-3413257 


•003921569 


256 


65536 


16777216 


16-0000000 


6-3496042 


•003906250 


257 


66049 


16974593 


16-0312195 


6-3578611 


•003891051 


258 


66564 


17173512 


16-0623784 


6-3660968 


•003875969 


259 


67081 


17373979 


16-0934769 


6-3743111 


•003861004 


260 


67600 


17576000 


16-1245155 


6-3825043 


•003846154 


261 


68121 


17779581 


16-1554944 


6-3906765 


•003831418 


262 


68644 


17984728 


16-1864141 


6-3988279 


•003816794 


263 


69169 


18191447 


16-2172747 


6-4069585 


•003802281 


264 


69696 


18399744 


16-2480768 


6-4150087 


•003787879 


265 


70225 


18609625 


16-2788206 


6-4231583 


•003773585 


266 


70756 


18821096 


16-3095064 


6-4312276 


•003759398 


267 


71289 


19034163 


16-3401346 


6-4392767 


•003745318 


268 


71824 


19248832 


16-3707055 


6-4473057 


•003731343 


269 


72361 


19465109 


16-4012195 


6-4553148 


•003717472 


270 


72900 


19683000 


16-4316767 


6-4633041 


•003703704 


271 


73441 


19902511 


16-4620776 


6-4712736 


•003690037 


272 


73984 


20123643 


16-4924225 


6-4792236 


•003676471 


273 


74529 


20346417 


16-5227116 


6-4871541 


•003663004 


274 


75076 


20570824 


16-5529454 


6-4950653 


•003649635 


275 


75625 


20796875 


16-5831240 


6-5029572 


•003636364 


276 


76176 


21024576 


16-6132477 


6-5108300 


•003623188 


277 


76729 


21253933 


16-6433170 


6-5186839 


•003610108 


278 


77284 


21484952 


16-6783320 


6-5265189 


•003597122 


279 


77841 


21717639 


16-7032931 


6-5343351 


•003584229 


280 


78400 


21952000 


16-7332005 


6-5421326 


•003571429 


281 


78961 


22188041 


16-7630546 


6-5499116 


•003558719 


282 


79524 


22425768 


16-7928556 


6-5576722 


•003546099 


283 


80089 


22665187 


16-8226038 


6-5654144 


•003533569 


284 


80656 


22906304 


16-8522995 


6-5731385 


•003522127 


285 


81225 


23149125 


16-8819430 


6-5808443 


•003508772 


286 


81796 


23393656 


16-9115345 


6-5885323 


•003496503 


.287 


82369 


23639903 


16-9410743 


6-5962023 


•003484321 


288 


82944 


23887872 


16-9705627 


6-6038545 


•003472222 


289 


83521 


24137569 


17-0000000 


6-6114890 


■003460208 


290 


84100 


24389000 


17-0293864 


6-6191060 


•003448276 


291 


84681 


24642171 


17-0587221 


6-6267054 


•003436426 


292 


85264 


24897088 


17-0880075 


6-6342874 


•003424658 


293 


85849 


25153757 


17-1172428 


6-6418522 


•003412969 


294 


86436 


25412184 


17-1464282 


6-6493998 


•003401361 


295 


87025 


25672375 


17-1755640 


6-6569302 


•003389831 


296 


87616 


25934836 


17-2046505 


6-6644437 


•003378378 


297 


88209 


26198073 


17-2336879 


6-6719403 


•003367003 


298 


88804 


26463592 


17-2626765 


6-6794200 


•003355705 


299 


89401 


26730899 


17-2916165 


6-6868831 


•003344482 


SOO 


90000 


27000000 


17-3205081 


6-6943295 


•003333333 


301 


90601 


27270901 


17-3493516 


6-7017593 


•003322259 


302 


91204 


27543608 


17-3781472 


6-7091729 


•003311258 


303 


91809 


27818127 


17-4068952 


6-7165700 


•003301330 


304 


92416 


28094464 


17-4355958 


6-7239508 


-003289474 


805 


93025 


28372625 


17-4642492 


6-7313155 


-003278689 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 105 



Numlier. 


Squares. 


Cubes. 


Square Koots. 


Cube Roots. 


Reciprocals. 


306 


93636 


28652616 


17-4928557 


6-7386641 


•003267974 


307 


94249 


28934443 


17-5214155 


6-7459967 


•003257329 


308 


94864 


29218112 


17-5499288 


6-7533134 


•003246753 


309 


95481 


29503609 


17-5783958 


6-7606143 


•003236246 


310 


96100 


29791000 


17-6068169 


6-7678995 


•003225806 


311 


96721 


30080231 


17-6351921 


6-7751690 


•003215434 


312 


97344 


30371328 


17-6635217 


6-7824229 


•003205128 


313 


97969 


30664297 


17-6918060 


6-7896613 


•003194888 


314 


98596 


30959144 


17-7200451 


6-7968844 


•003184713 


315 


99225 


31255875 


17-7482393 


6-8040921 


•003174603 


316 


99856 


31554496 


17-7763888 


6-8112847 


•003164557 


317 


100489 


31855013 


17-8044938 


6-8184620 


•003154574 


318 


101124 


32157432 


17-8325545 


6-8256242 


•003144654 


319 


101761 


32461759 


17-8(305711 


6-8327714 


• •003134796 


320 


102400 


32768000 


17-8885438 


6-8399037 


•003125000 


321 


103041 


33076161 


17-9164729 


6-8470213 


•003115285 


322 


103684 


33386248 


17-9443584 


6-8541240 


•003105590 


323 


104329 


33698267 


17-9722008 


6-8612120 


•003095975 


324 


104976 


34012224 


18-0000000 


6-8682855 


•003086420 


325 


105625 


34328125 


18-0277564 


6-8753433 


•003076923 


326 


106276 


34645976 


18-0554701 


6-8823888 


-003067485 


327 


106929 


34965783 


18-0831413 


6-8894188 


•003058104 


328 


107584 


35287552 


18-1107703 


6-8964345 


•003048780 


329 


108241 


35611289 


18-1383571 


6-9034359 


•003039514 


330 


108900 


35937000 


18-1659021 


6-9104232 


•003030303 


331 


109561 


36264691 


18-1934054 


6-9173964 


•003021148 


332 


110224 


36594368 


18-2208672 


6-9243556 


•003012048 


333 


110889 


36926037 


18-2482876 


6-9313088 


•003003003 


334 


111556 


37259704 


18-2756669 


6-9382321 


•002994012 


335 


112225 


37595375 


18-3030052 


6-9451496 


•002985075 


336 


112896 


37933056 


18-3303028 


6-9520533 


•002970190 


337 


113569 


38272753 


18-3575598 


6-9589434 


•002907359 


338 


114244 


38614472 


18-3847763 


6-9658198 


•002958580 


339 


114921 


38958219 


18-4119526 


6-9726826 


•002949853 


340 


115600 


39304000 


18-4390889 


6-9795321 


•002941176 


341 


116281 


39651821 


18-4661853 


6-9863681 


•002932551 


342 


116964 


40001688 


18-4932420 


6-9931906 


•002923977 


343 


117649 


40353607 


18-5202592 


7-0000000 


•002915452 


344 


118336 


40707584 


18-5472370 


7-0067962 


•002906977 


345 


119025 


41063625 


18-5741756 


7-0135791 


•002898551 


346 


119716 


41421736 


18-6010752 


7-0203490 


•002890173 


347 


120409 


41781923 


18-6279360 


7-0271058 


•002881844 


348 


121104 


42144192 


18-6547581 


7-0338497 


•002873563 


349 


121801 


42508549 


18-6815417 


7-0405860 


•002865330 


350 


122500 


42875000 


18-7082869 


7-0472987 


•002857143 


351 


123201 


43243551 


18-7349940 


7-0540041 


•002849003 


352 


123904 


43614208 


18-7616630 


7-0606967 


•002840909 


353 


124609 


43986977 


18-7882942 


7-0673767 


•002832861 


354 


125316 


44361864 


18-8148877 


7-0740440 


•002824859 


355 


126025 


44738875 


18-8414437 


7-0806988 


•002816901 


356 


126736 


45118016 


18-8679623 


7-0873411 


•002808989 


357 


127449 


45499293 


18-8944436 


7-0939709 


•002801120 


358 


128164 


45882712 


18-9208879 


7-1005885 


•002793296 


359 


128881 


46268279 


18-9472953 


7-1071937 


•002785515 


360 


129600 


46656000 


18-9736660 


7-1137866 


•002777778 


361 


130321 


47045831 


19-0000000 


7-1203674 


•002770083 


362 


131044 


47437928 


19-0262976 


7-1269360 


•002762431 


363 


131769 


47832147 


19-0525589 


7-1334925 


•002754821 


364 


132496 


' 48228544 


19-0787840 


7-1400370 


•002747253 


365 


133225 


48627125 


19-1049732 


7-1465695 


•002739726 


366 


133956 


49027896 


19-1311265 


7-1530901 


•002732240 


367 


134689 


49430863 


19-1572441 


7-1595988 


•002724796 



106 



THE PRACTICAL MODEL CALCULATOR. 



NiTmber. 


Squares. 


Cubes. 


Square Roots. 


Cube Eoots. 


Reciprocals. 


368 


135424 


49836032 


19-1833261 


7-1660957 


•002717391 


369 


136161 


50243409 


19-2093727 


7-1725809 


•002710027 


370 


136900 


60653000 


19-2353841 


7-1790544 


•002702703 


871 


137641 


51064811 


19-2613603 


7-1855162 


•002695418 


372 


138384 


61478848 


19-2873015 


7-1919663 


•002688172 


373 


139129 


51895117 


19-3132079 


7-1984050 


•002680905 


374 


139876 


62313624 


19-3390796 


7-2048322 


•002073797 


375 


140625 


62734375 


19-3649167 


7-2112479 


•0020G6667 


376 


141376 


53157376 


19-3907194 


7-2176522 


•002659574 


377 


142129 


63582633 


19-4164878 


7-2240450 


•002652520 


378 


142884 


64010152 


19-4422221 


7-2304268 


•002645503 


379 


143641 


54439939 


19-4679223 


7-2367972 


•002638521 


380 


144400 


64872000 


19-4935887 


7-2431565 


•002631579 


381 


145161 


55306341 


19-5192213 


7-2495045 


-002624672 


382 


145924 


65742968 


19-5448203 


7-2558415 


-002G17801 


383 


146689 


56181887 


19-5703858 


7-2621675 


-002610966 " 


384 


147456 


66623104 


19-5959179 


7-2684824 


•002604167 


385 


148225 


67066625 


19-6214169 


7-2747864 


•002597403 


386 


148996 


67512456 


19-6468827 


7-2810794 


•002590674 


387 


149769 


57960603 


19-6723156 


7-2873617 


•002583979 


388 


150544 


68411072 


19-6977156 


7-2936330 


-002577320 


389 


151321 


68863869 


19-7230829 


7-2998936 


•002570694 


390 


152100 


69319000 


19-7484177 


7-3061436 


•002504103 


391 


152881 


59776471 


19-7737199 


7-3123828 


•002557545 


392 


153664 


60236288 


19-7989899 


7-3186114 


•002551020 


393 


154449 


60698457 


19-8242276 


7-3248295 


•002544529 


394 


155236 


61162984 


19-8494332 


7-3310369 


•002538071 


395 


156025 


61629875 


19-8746069 


7-3372339 


•002531646 


396 


156816 


62099136 


19-8997487 


7-3434205 


•002525253 


397 


157609 


62570773 


19-9248588 


7-3495966 


•002518892 


398 


158404 


63044792 


19-9499373 


7-3557624 


•002512563 


399 


159201 


63521199 


19-9749844 


7-3619178 


•002506266 


400 


160000 


64000000 


20-0000000 


7-3680630 


-002500000 


401 


160801 


64481201 


20-0249844 


7-3741979 


•002493766 


402 


161604 


64964808 


20-0499377 


7-3803227 


•0024875G2 


403 


162409 


65450827 


20-0748599 


7-3864373 


•002481390 


404 


163216 


65939264 


20-0997512 


7-3925418 


•002475248 


405 


164025 


66430125 


20-1246118 


7-3986363 


•002469136 


406 


164836 


66923416 


20-1404417 


7-4047206 


•002463054 


407 


165649 


67419143 


20-1742410 


7-4107950 


•002457002 


408 


166464 


67917312 


20-1990099 


7-4168595 


•002450980 


409 


167281 


68417929 


20-2237484 


7-4229142 


•002444988 


410 


168100 


68921000 


20-2484567 


7-4289589 


•002439024 


411 


168921 


69426531 


20-2731349 


7-4349938 


•002433090 


412 


169744 


69934528 


20-2977831 


7-4410189 


•002427184 


413 


170569 


70444997 


20-3224014 


7-4470343 


•002421308 


414 


171396 


70957944 


20-3469899 


7-4530399 


•002415459 


416 


172225 


71473375 


20-3715488 


7-4690359 


•002409639 


416 


173056 


71991296 


20-3960781 


7-4650223 


•002406846 


417 


173889 


72511713 


20-4205779 


7-4709991 


•002398082 


418 


174724 


73034632 


20-4450483 


7-4769664 


•002392344 


419 


175561 


73560059 


20-4694895 


7-4829242 


•002386635 


420 


176400 


74088000 


20-4939015 


7-4888724 


•002380952 


421 


177241 


74618461 


20-5182845 


7-4948113 


•002375297 


422 


178084 


75151448 


20-5426386 


7-6007406 


•002369668 


423 


178929 


75686967 


20-5669638 


7-5066607 


•002364066 


424 


179776 


76225024 


20-5912603 


7-5125715 


•002358491 


425 


180625 


76765625 


20-6155281 


7-5184730 


•002352941 


426 


181476 


77308776 


20-6397674 


7-6243652 


•002347418 


427 


182329 


77854483 


20-6639783 


7-5302482 


•002341920 


428 


183184 


78402752 


20-6881609 


7-6361221 


•002336449 


429 


184041 


78953589 


20-7123152 


7-6419867 


•002331002 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 107 



Number. 


Squares. 


Cubes. 


Square Koots. 


Cube Roots. 


Reciprocals. 


430 


184900 


79507000 


20-7364414 


7-5478423 


•002325581 


431 


185761 


80062991 


20-7605395 


7-5536888 


-002320186 


432 


186624 


80621568 


20-7846097 


7-5595263 


-002314815 


433 


187489 


81182737 


20-8086520 


7-5653548 


•002309469 


434 


188356 


81746504 


20-8326667 


7-5711743 


•002304147 


435 


189225 


82312875 


20-8566536 


7-5769849 


•002298851 


436 


190096 


82881856 


20-8806130 


7-6827865 


•002293578 


437 


190969 


83453453 


20-9045450 


7-5885793 


-002288330 


438 


191844 


84027672 


20-9284495 


7-5943633 


•002283105 


439 


192721 


84604519 


20-9523268 


7-6001385 


•002277904 


440 


193600 


85184000 


20-9761770 


7-6059049 


•002272727 


441 


194481 


85766121 


21-0000000 


7-6116626 


•002267574 


442 


195364 


86350888 


21-0237960 


7-6174116 


•002262443 


443 


196249 


86938307 


21-0475652 


7-6231619 


•002257336 


444 


197136 


87528384 


21-0713075 


7-6288837 


•002262252 


445 


198025 


88121125 


21-0950231 


7-6346067 


•002247191 


446 


198916 


88716536 


21-1187121 


7-6403213 


•002242152 


447 


199809 


89314623 


21-1423745 


7-6460272 


•002237136 


448 


200704 


89915392 


21-1660105 


7-6517247 


•002232143 


449 


201601 


90518849 


21-1896201 


7-6574138 


-002227171 


450 


202500 


91125000 


21-2132034 


7-6630943 


-002222222 


451 


203401 


91733851 


21-2367606 


7-6687665 


•002217295 


452 


204304 


92345408 


21-2602916 


7-6744303 


•002212389 


453 


205209 


92959677 


21-2837967 


7-6800857 


•002207606 


454 


206116 


93576664 


21-3072758 


7-6857328 


•002202643 


455 


207025 


94196375 


21-3307290 


7-6913717 


•002197802 


456 


207936 


94818816 


21-3541565 


7-6970023 


•002192982 


457 


208849 


95443993 


21-3775583 


7-7026246 


•002188184 


458 


209764 


96071912 


21-4009346 


7-7082388 


•002183406 


459 


210681 


96702579 


21-4242853 


7-7188448 


•002178649 


460 


211600 


97336000 


21-4476106 


7-7194426 


•002173913 


461 


212521 


97972181 


21-4709106 


7-7250325 


-002169197 


462 


213444 


98611128 


21-4941853 


7-7306141 


•002164502 


463 


214369 


99252847 


21-5174348 


7-7361877 


•002159827 


464 


215296 


99897344 


21-5406592 


7-7417532 


•002155172 


465 


216225 


100544625 


21-5638587 


7-7473109 


•002150538 


466 


217156 


101194696 


21-5870331 


7-7528606 


•002145923 


467 


218089 


101847563 


21-6101828 


7-7584023 


•002141328 


468 


219024 


102503232 


21-6333077 


7-7639361 


•002136752 


469 


219961 


103161709 


21-6564078 


7-7694620 


•002132196 


470 


220900 


103823000 


21-6794834 


7-7749801 


•002127660 


471 


221841 


104487111 


21-7025344 


7-7804904 


•002123142 


472 


222784 


105154048 


21-7255610 


7-7859928 


•002118644 


473 


223729 


105828817 


21-7485632 


7-7914875 


•002114165 


474 


224676 


106496424 


21-7715411 


7-7969745 


•002109705 


475 


225625 


107171875 


21-7944947 


7-8024538 


•002105263 


476 


226576 


107850176 


21-8174212 


7-8079254 


•002100840 


477 


227529 


108531333 


21-8403297 


7-8133892 


•002096486 


478 


228484 


109215352 


21-8632111 


7-8188456 


-002092050 


479 


229441 


109902239 


21-8860686 


7-8242942 


•002087683 


480 


230400 


110592000 


21-9089023 


7-8297353 


•002083333 


481 


231361 


111284641 


21-9317122 


7-8351688 


•002079002 


482 


232324 


111980168 


21-9544984 


7-8405949 


-002074689 


483 


233289 


112678587 


21-9772610 


7-8460134 


-002070393 


484 


234256 


113379904 


22-0000000 


7-8514244 


•002066116 


485 


235225 


114084125 


22-0227155 


7-8568281 


•002061856 


486 


236196 


114791256 


22-0454077 


7-8622242 


•002057613 


487 


237169 


115501303 


22-0680765 


7-8676130 


•002053388 


488 


238144 


116211272 


22-0907220 


7-8729944 


•002049180 


489 


239121 


116930169 


22-1133444 


7-8783684 


•002044990 


490 


240100 


117649000 


22-1359436 


7-8837352 


•002040816 


491 


241081 


118370771 


22-1585198 


7-8890946 


-002036660 



108 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


Square Eoots. 


Cube Eoots. 


Keciprocals. 


492 


242064 


119095488 


22-1810730 


7-8944468 


•002032520 


493 


243049 


119823157 


22-2036033 


7-8997917 


•002028398 


494 


244036 


120553784 


22-2261108 


7-9051294 


•002024291 


495 


245025 


121287375 


22-2485955 


7-9104599 


•002020202 


496 


246016 


122023936 


22-2710575 


7-9157832 


•002016129 


497 


247009 


122763473 


22-2934968 


7-9210994 


•002012072 


498 


248004 


123505992 


22-3159136 


7-9264085 


•002008032 


499 


249001 


124251499 


22-3383079 


7-9317104 


•002004008 


500 


250000 


125000000 


22-3606798 


7-9370053 


•002000000 


501 


251001 


125751501 


22-3830293 


7-9422931 


•001996008 


502 


252004 


126506008 


22-4053565 


7-9475739 


•001992032 


503 


253009 


127263527 


22-4276615 


7-9528477 


•001988072 


504 


254016 


128024064 


22-4499443 


7-9581144 


•001984127 


505 


255025 


128787625 


22-4722051 


7-9633743 


•001980198 


506 


256036 


129554216 


22-4944438 


7-9686271 


•001976285 


507 


257049 


130323843 


22-5166605 


7-9738731 


•001972387 


508 


258064 


131096512 


22-5388553 


7-9791122 


•001968504 


509 


259081 


131872229 


22-5610283 


7-9843444 


•001964637 


1 610 


260100 


132651000 


22-5831796 


7-9895697 


•001960784 


511 


261121 


133432831 


22-6053091 


7-9947883 


•001956947 


512 


262144 


134217728 


22-6274170 


8-0000000 


•001953125 


513 


263169 


135005697 


22-6495033 


8-0052049 


•001949318 


514 


264196 


135796744 


22-6715681 


8-0104032 


•001945525 


515 


265225 


136590875 


22-6936114 


8-0155946 


•001941748 


516 


266256 


137388096 


22-71.56334 


8-0207794 


•001937984 


617 


267289 


138188413 


22-7376341 


8-0259574 


•001934236 


518 


268324 


138991832 


22-7596134 


8-0311287 


•001930502 


519 


269361 


139798359 


22-7815715 


8-0362935 


•001926782 


620 


270400 


140608000 


22-8035085 


8-0414515 


•001923077 


521 


271411 


141420761 


22-8254244 


8-0466030 


•00191938G 


522 


272484 


142236648 


22-8473193 


8-0517479 


•001915709 


523 


273529 


143055667 


22-8691933 


8-0568862 


•001912046 


524 


274576 


143877824 


22-8910463 


8-0620180 


•001908397 


525 


275625 


144703125 


22-9128785 


8-0671432 


-001904762 


526 


276676 


145531576 


22-9346899 


8-0722620 


•001901141 


627 


277729 


146363183 


22-9564806 


8-0773743 


•001897533 


528 


278784 


147197952 


22-9782506 


8-0824800 


•001893939 


529 


279841 


148035889 


23-0000000 


8-0875794 


•001890359 


630 


280900 


148877001 


23-0217289 


8-0926723 


•001886792 


531 


281961 


149721291 


23-0434372 


8-0977589 


•001883239 


532 


283024 


150568768 


23-0651252 


8-1028390 


•001879699 


533 


284089 


151419437 


23-0867928 


8-1079128 


•001876173 


634 


285156 


152273304 


23-1084400 


8-1129803 


•001872659 


635 


286225 


153130375 


23-1300670 


8-1180414 


•001869159 


536 


287296 


153990656 


23-1516738 


8-1230962 


•001865672 


537 


288369 


154854153 


23-1732605 


8-1281447 


•001862197 


638 


289444 


155720872 


23-1948270 


8-1331870 


•001858736 


539 


290521 


156590819 


23-2163735 


8-1382230 


•001855288 


540 


291600 


157464000 


23-2379001 


8-1432529 


•001851852 


541 


292681 


158340421 


23-2594067 


8-1482765 


•001848429 


542 


293764 


159220088 


23-2808935 


8-1532939 


•001845018 


543 


294849 


160103007 


23-3023604 


8-1583051 


•001841621 


544 


295936 


160989184 


23-3238076 


8-1633102 


•001838235 


545 


297025 


161878625 


23-3452351 


8-1683092 


•001834862 


546 


298116 


162771336 


23-3666429 


8-1733020 


•001831502 


547 


299209 


163667323 


23-3880311 


8-1782888 


•001828154 


648 


300304 


164566592 


23-4093998 


8-1832695 


•001824818 


549 


301401 


165469149 


23-4307490 


8-1882441 


•001821494 


550 


302500 


166375000 


23-4520788 


8-1932127 


•001818182 


551 


303601 


167284151 


23-4733892 


8-1981753 


•001814882 


552 


304704 


168196608 


23-4946802 


8-2031319 


-001811594 


653 


305809 


169112377 


23-5159520 


8-2080825 


•001808318 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 109 



Number. 


Sqoares. 


Cubes. 


Square Eoots. 


Cube Roots. 


Keciproeals. 


654 


306916 


170031464 


23-5372046 


8-2130271 


•001805054 


655 


308025 


170953875 


23-5584380 


8-2179657 


-001801802 


656 


309136 


171879616 


23-5796522 


8-2228985 


-001798561 


657 


310249 


172808693 


23-6008474 


8-2278254 


•001795332 


658 


311364 


173741112 


23-6220236 


8-2327463 


-001792115 


559 


312481 


174676879 


23-6431808 


8-2376614 


•001788909 


660 


313600 


175616000 


23-6643191 


8-2425706 


•001785714 


661 


314721 


176558481 


23-6854386 


8-2474740 


•001782531 


562 


315844 


177504328 


23-7065392 


8-2523715 


•001779359 


663 


316969 


178453547 


23-7276210 


8-2572635 


•001776199 


664 


318096 


179406144 


23-7486842 


8-2621492 


•001773050 


665 


319225 


180362125 


23-7697286 


8-2670294 


•001769912 


666 


320356 


181321496 


23-7907545 


8-2719039 


•001766784 


567 


321489 


182284263 


23-8117618 


8-2767726 


•001763668 


668 


322624 


183250432 


23-8327506 


8-2816255 


•001760563 


669 


323761 


184220009 


23-8537209 


8-2864928 


•001757469 


670 


324900 


185193000 


23-8746728 


8-2913444 


■001754386 


671 


326041 


186169411 


23-8956063 


8-2961903 


•001751313 


572 


327184 


187149248 


23-9165215 


8-3010304 


•001748252 


673 


328329 


188132517 


23-9374184 


8-3058651 


•001745201 


574 


329476 


189119224 


23-9582971 


8-3106941 


-001742160 


675 


330625 


190109375 


23-9791576 


8-3155175 


•001739130 


676 


331776 


191102976 


24-0000000 


8-3203353 


•001736111 


677 


332927 


192100033 


24-0208243 


8-3251475 


•001733102 


578 


334084 


193100552 


24-0416306 


8-3299542 


•001730104 


679 


335241 


194104539 


24-0624188 


8-3347553 


-001727116 


680 


336400 


195112000 


24-0831891 


8-3395509 


-001724138 


581 


337561 


196122941 


24-1039416 


8-3443410 


•001721170 


582 


338724 


197137368 


24-1246762 


8-3491256 


•001718213 


583 


339889 


198155287 


24-1453929 


8-3539047 


•001715266 


584 


341056 


199176704 


24-1660919 


8-3586784 


•001712329 


585 


342225 


200201625 


24-1867732 


8-3634466 


-001709402 


586 


343396 


201230056 


24-2074369 


8-3682095 


•001706485 


587 


344569 


202262003 


24-2280829 


8-3729668 


•001703578 


588 


345744 


203297472 


24-2487113 


8-3777188 


•001700680 


689 


346921 


204336469 


24-2693222 


8-3824653 


•001697793 


690 


348100 


205379000 


24-2899156 


8-3872065 


•001694915 


591 


349281 


206425071 


24-3104996 


8-3919428 


•001692047 


692 


350464 


207474688 


24-3310501 


8-3966729 


•001689189 


693 


351649 


208527857 


24-3515913 


8-4013981 


•001686341 


694 


352836 


209584584 


24-3721152 


8-4061180 


•001683502 


695 


354025 


210644875 


24-3926218 


8-4108326 


•001680672 


696 


355216 


211708736 


24-4131112 


8-4155419 


•001677852 


597 


356409 


212776173 


24-4335834 


8-4202460 


•001675042 


598 


357604 


213847192 


24-4540385 


8-4249448 


•001672241 


599 


358801 


214921799 


24-4744765 


8-4296383 


•001669449 


600 


360000 


216000000 


24-4948974 


8-4343267 


•001666667 


601 


361201 


217081801 


24-5153013 


8-4390098 


•001663894 


602 


362404 


218167208 


24-5356883 


8-4436877 


•001661130 


603 


363609 


219256227 


24-5560583 


8-4483605 


•001658375 


604 


364816 


220348864 


24-5764115 


8-4530281 


•001655629 


605 


366025 


221445125 


24-5967478 


8-4576906 


•001652893 


606 


367236 


222545016 


24-6170673 


8-4623479 


•001650165 


607 


368449 


223648543 


24-6373700 


8-4670001 


•001647446 


608 


369664 


224755712 


24-6576560 


8-4716471 


•001644737 


609 


370881 


225866529 


24-6779254 


8-4762892 


•001642036 


610 


372100 


226981000 


24-6981781 


8-4809261 


•001639344 


611 


373321 


228099131 


24-7184142 


8-4855579 


•001636661 


612 


374544 


229220928 


24-7386838 


8-4901848 


-001633987 


613 


375769 


230346397 


24-7588368 


8-4948065 


-001631321 


614 


376996 


231475544 


24-7790234 


8-4994233 


•001628664 


615 


378225 


232608375 


24-7991935 


8-5040350 


•001626016 



no 



THE PRACTICAL MODEL CALCULATOE. 



Number. 


Squares, j 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


616 


379456 


233744896 


24-8193473 


8-.5086417 


•001623377 


617 


380689 


234885113 


24-8394847 


8-5132435 


-001620746 


618 


381924 


236029032 


24-8596058 


8-5178403 


■001618123 


619 


383161 


237176659 


24-8797106 


8--5224331 


•001615-509 


620 


384400 


238328000 


24-8997992 


8-5270189 


•001612903 


621 


385641 


239483061 


24-9198716 


8-5316009 


•001610306 


622 


386884 


240641848 


24-9399278 


8-5361780 


•001607717 


623 


388129 


241804367 


24-9599679 


8-5407501 


•001605136 


624 


389376 


242970624 


24-9799920 


8-5453173 


•001602564 


625 


390625 


244140625 


25-0000000 


8-5498797 


•001600000 


626 


391876 


245134376 


25-0199920 


8-5544372 


•001597444 


627 


393129 


246491883 


25-0399681 


8-5589899 


•001594896 


628 


394384 


247673152 


25-0599282 


8-563-5377 


•001592357 


629 


395641 


248858189 


25-0798724 


8-5680807 


•001589825 


630 


396900 


250047000 


25-0998008 


8-5726189 


•001587302 


631 


398161 


251239591 


25-1197134 


8-5771523 


•001584786 


632 


399424 


252435968 


25-1396102 


8-5816809 


•001582278 


633 


400689 


253636137 


25-1594913 


8-5862247 


•001579779 


634 


401956 


254840104 


25-1793566 


8-5907238 


•001577287 


635 


403225 


256047875 


25-1992063 


8-5952380 


•001574803 


636 


404496 


257259456 


25-2190404 


8-5997476 


•001572327 


637 


405769 


258474853 


25-2388589 


8-6042-525 


•001569859 


638 


407044 


259694072 


25-2586619 


8-6087526 


•001567398 


639 


408321 


260917119 


25-2784493 


8-6132480 


•001564945 


640 


409600 


262144000 


25-2982213 


8-6177388 


•001562500 


641 


410881 


263374721 


25-3179778 


8-6222248 


•001560062 


642 


412164 


264609288 


25-3377189 


8-6267063 


•001557632 


643 


413449 


265847707 


23-3574447 


8-6311830 


•001555210 


644 


414736 


267089984 


25-37715-51 


8-6356551 


•001552795 


645 


416125 


268336125 


25-3968-502 


8-6401226 


•001550388 


646 


417316 


269585136 


25-4165302 


8-6445855 


•001547988 


647 


418609 


270840023 


25-4361947 


8-6490437 


•001545595 


648 


419904 


272097792 


25-4558441 


8-6534974 


•001543210 


649 


421201 


273359449 


25-4754784 


8-6579465 


•001540832 


650 


422500 


274625000 


25-4950976 


8-6623911 


•001538462 


651 


423801 


275894451 


25-5147013 


8-6668310 


•001536098 


652 


425104 


277167808 


25-5342907 


8-6712665 


•001533742 


653 


426409 


278445077 


25-5538647 


8-6756974 


•001531394 


654 


427716 


279726264 


25-5734237 


8-6801237 


•0015290-52 


655 


429025 


281011375 


25-5929678 


8-6845456 


•001526718 


656 


430336 


282300416 


25-6124969 


8-6889630 


•001524390 


657 


431639 


283593393 


25-6320112 


8-6933759 


•001522070 


658 


432964 


284890312 


25-6-515107 


8-6977843 


•001519751 


659 


434281 


286191179 


25-6709953 


8-7021882 


•001517451 


660 


435600 


287496000 


25-6904652 


8-7065877 


•001515152 


^1 


436921 


288804781 


25-7099203 


8-7109827 


•0015128.59 


662 


438244 


290117528 


25-7293607 


8-71-53734 


•001510574 


663 


439569 


291434247 


25-7487864 


8-7197596 


•001508296 


664 


440896 


292754944 


25-7681975 


8-7241414 


•001506024 


665 


442225 


294079625 


25-7875939 


8-728-5187 


•001503759 


666 


443556 


295408296 


25-8069758 


8-7328918 


•001501502 


667 


444899 


296740963 


25-8263431 


8-7372604 


•001499250 


668 


446224 


298077632 


25-8456960 


8-7416246 


•001497006 


669 


447561 


299418309 


25-8650343 


8-7459846 


•001494768 


670 


448900 


300763000 


25-8843582 


8-7503401 


•001492537 


671 


450241 


302111711 


25-9036677 


8-7-546913 


•001490313 


672 


451584 


303464448 


25-9229628 


8-759038-5 


•001488095 


673 


452929 


304821217 


25-9422435 


8-7633809 


•001485884 


674 


454276 


306182024 


25-9615100 


8-7677192 


•001483680 


675 


455625 


307546875 


25-9807621 


8-7720532 


•001481481 


676 


456976 


308915776 


26-0000000 


8-7763830 


•001479290 


677 


458329 


310288733 


26-0192237 


8-7807084 


•001477105 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



Ill 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


678 


459684 


311665752 


26-0384331 


8-7850296 


•001474926 


679 


461041 


313046839 


26-0576284 


8-7893466 


•001472754 


680 


462400 


314432000 


26-0768096 


8-7936593 


•001470588 


681 


463761 


315821241 


26-0959767 


8-7979679 


•001468429 


682 


465124 


317214568 


26-1151297 


8-8022721 


-001466276 


683 


466489 


318611987 


26-1342687 


8-8065722 


•001464129 


684 


467856 


320013504 


26-1533937 


8-8108681 


•001461988 


685 


469225 


321419125 


26-1725047 


8-8151598 


•001459854 


686 


470596 


322828856 


26-1916017 


8-8194474 


•001457726 


687 


471969 


324242703 


26-2106848 


8-8237307 


•001455604 


688 


473344 


325660672 


26-2297541 


8-8280099 


•001453488 


689 


474721 


327082769 


26-2488095 


8-8322850 


•001451379 


690 


476100 


328509000 


26-2678511 


8-8365559 


•001449275 


691 


477481 


329939371 


26-2868789 


8-8408227 


•001447178 


692 


478864 


331373888 


26-3058929 


8-8450854 


•001445087 


693 


480249 


332812557 


26-3248932 


8-8493440 


•001443001 


694 


481636 


334255384 


26-3438797 


8-8535985 


•001440922 


695 


483025 


335702375 


26-3628527 


8-8578489 


•001438849 


696 


484416 


337153536 


26-3818119 


8-8620952 


•001436782 


697 


485809 


338608873 


26-4007576 


8-8663375 


•001434720 


698 


487204 


340068392 


26-4196896 


8-8705757 


•001432665 


699 


488601 


341532099 


26-4386081 


8-8748099 


•001430615 


700 


490000 


343000000 


20-4575131 


8-8790400 


•001428571 


701 


491401 


344472101 


26-4764046 


8-8832661 


•001426534 


702 


492804 


345948408 


26-4952826 


8-8874882 


•001424501 


703 


494209 


347428927 


26-5141472 


8-8917063 


•001422475 


704 


495616 


348913664 


26-5329983 


8-8959204 


•001420455 


705 


497025 


350402625 


26-5518861 


8-9001304 


•001418440 


706 


498436 


351895816 


26-5706605 


8-9043366 


•001416431 


707 


499849 


353393243 


26-5894716 


8-9085387 


•001414427 


708 


501264 


354894912 


26-6082694 


8-9127369 


•001412429 


709 


502681 


356400829 


26-6270539 


8-9169311 


•001410437 


710 


504100 


357911000 


26-6458252 


8-9211214 


•001408451 


711 


505521 


359425431 


26-6645833 


8-9253078 


•001406470 


712 


506944 


360944128 


26-6833281 


8-9294902 


•001404494 


713 


508369 


362467097 


26-7020598 


8-9336687 


•001402525 


714 


509796 


363994344 


26-7207784 


8-9378433 


•001400560 


715 


511225 


365525875 


26-7394839 


8-9420140 


•001398601 


716 


512656 


367061696 


26-7581763 


8-9461809 


•001396648 


717 


514089 


368601813 


26-7768557 


8-9503438 


•001394700 


718 


515524 


370146232 


26-7955220 


8-9545029 


•001392758 


719 


516961 


371694959 


26-8141754 


8-9586581 


•001390821 


720 


518400 


373248000 


20-8328157 


8-9628095 


•001388889 


721 


519841 


374805361 


26-8514432 


8-9669570 


•001386963 


722 


521284 


376367048 


26-8700577 


8-9711007 


•001385042 


723 


522729 


377933067 


26-8886593 


8-9752406 


•001383126 


724 


524176 


379503424 


26-9072481 


8-9793766 


•001381215 


725 


525625 


381078125 


26-9258240 


8-9835089 


•001379310 


726 


527076 


382657176 


26-9443872 


8-9876373 


•001377410 


727 


528529 


384240583 


26-9629375 


8-9917620 


-001375516 


728 


529984 


385828352 


26-9814751 


8-9958899 


-001373626 | 


729 


531441 


387420489 


27-0000000 


9-0000000 


•001371742 


730 


532900 


389017000 


27-0185122 


9-0041134 


•001369863 


731 


534361 


390617891 


27-0370117 


9-0082229 


•001367989 


732 


535824 


392223168 


27-0554985 


9-0123288 


•001366120 


733 


537289 


393832837 


27-0739727 


9-0164309 


•001364256 


734 


538756 


395446904 


27-0924344 


9-0205293 


•001362398 


735 


540225 


397065375 


27-1108834 


9-0246239 


•001360544 


73G 


541696 


398688256 


27-1293199 


9-0287149 


•001358696 


737 


543169 


400315553 


27-1477149 


9-0328021 


•001356852 


738 


541644 


401947272 


27-1661554 


9-0368857 


•001355014 


739 


546121 


403583419 


27-1845544 


9-0409655 


•001353180 



112 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


740 


547600 


405224000 


27-2029140 


9-0450419 


•001351351 


741 


549801 


406869021 


27-2213152 


9-0491142 


•001349528 


742 


550564 


408518488 


27-2396769 


9-0531831 


•001347709 


743 


552049 


410172407 


27-2580263 


9-0572482 


•001345895 


744 


553536 


411830784 


27-2763634 


9-0613098 


•001344086 


745 


555025 


413493625 


27-2946881 


9-0653677 


•001342282 


746 


556516 


415160936 


27-3130006 


9-0694220 


•001340483 


747 


558009 


416832723 


27-3313007 


9-0734726 


•001338688 


748 


659504 


418508992 


27-3495887 


9-0775197 


•001336898 


749 


561001 


420189749 


27-3678644 


9-0815631 


•001335113 


750 


562500 


421875000 


27-3861279 


9-0856030 


•001333333 


751 


564001 


423564751 


27-4043792 


9-0896352 


•001331558 


752 


565504 


425259008 


27-4226184 


9-0936719 


•001329787 


753 


567009 


426957777 


27-4408455 


9-0977010 


•001328021 


754 


568516 


428661064 


27-4590604 


9-1017265 


•001326260 


755 


570025 


430368875 


27-4772633 


9-1057485 


•001324503 


756 


571536 


432081216 


27-4954542 


9-1097669 


•001322751 


757 


573049 


433798093 


27-5136330 


9-1137818 


•001321004 


758 


574564 


435519512 


27-5317998 


9-1177931 


•001319261 


759 


576081 


437245479 


27-5499546 


9-1218010 


•001317523 


760 


577600 


438976000 


27-5680975 


9-1258053 


•001315789 


761 


579121 


440711081 


27-5862284 


9-1298061 


•001314060 


762 


580644 


442450728 


27-6043475 


9-1338034 


•001312336 


763 


582169 


444194947 


27-6224546 


9-1377971 


•001310616 


764 


583696 


445943744 


27-6405499 


9-1417874 


•001308901 


765 


585225 


447697125 


27-6586334 


9-1457742 


•001307190 


766 


586756 


449455096 


27-6767050 


9-1497576 


•001305483 


767 


588289 


451217663 


27-6947648 


9-1537375 


•001303781 


768 


589824 


452984832 


27-7128129 


9-1577139 


•001302083 


769 


591361 


454756609 


27-7308492 


9-1616869 


•001300390 


770 


592900 


456533000 


27-7488739 


9-1656565 


•001298701 


771 


594441 


458314011 


27-7668868 


9-1696225 


•001297017 


772 


595984 


460099648 


27-7848880 


9-1735852 


•001295337 


773 


597529 


461889917 


27-8028775 


9-1775445 


•001293661 


774 


599076 


463684824 


27-8208555 


9-1815003 


•001291990 


775 


600625 


465484375 


27-8388218 


9-1854527 


•001290323 


776 


602176 


467288576 


27-8567766 


9-1894018 


•001288660 


777 


603729 


469097433 


27-8747197 


9-1933474 


•001287001 


778 


605284 


470910952 


27-8926514 


9-1972897 


•001285347 


779 


606841 


472729139 


27-9105715 


9-2012286 


•001283697 


780 


608400 


474552000 


27-9284801 


9-2051641 


•001282051 


781 


609961 


476379541 


27-9463772 


9-2090962 


•001280410 


782 


611524 


478211768 


27-9642629 


9-2130250 


•001278772 


783 


613089 


480048687 


27-9821372 


9-2169505 


•001277139 


784 


614656 


481890304 


28-0000000 


9-2208726 


•001275510 


785 


616225 


483736625 


28-0178515 


9-2247914 


•001273885 


786 


617796 


485587656 


28-0356915 


9-2287068 


-001272265 


787 


619369 


487443403 


28-0535203 


9-2326189 


-001270648 


788 


620944 


489303872 


28-0713377 


9-2365277 


•001269036 


789 


622521 


491169069 


28-0891438 


9-2404333 


•001267427 


790 


624100 


493039000 


28-1069386 


9-2443355 


•001265823 


791 


625681 


494913671 


28-1247222 


9-2482344 


•001264223 


792 


627624 


496793088 


28-1424946 


8-2521300 


-001262626 


793 


628849 


498677257 


28-1602557 


9-2560224 


-001261034 


794 


630436 


500566184 


28-1780056 


9-2599114 


•001259446 


795 


632025 


502459875 


28-1957444 


9-2637973 


•001257862 


796 


633616 


504358336 


28-2134720 


9-2676798 


•001256281 


797 


635209 


506261573 


28-2311884 


9-2715592 


•001254705 


798 


636804 


508169592 


28-2488938 


9-2754352 


•001253133 


799 


638401 


510082399 


28-2665881 


9-2793081 


•001251364 


800 


640000 


512000000 


28-2842712 


9-2831777 


•001250000 


801 


641601 


513922401 


28-3019434 


9-2870444 


•001248439 



TABLE OF SQUARES, CUBES, SQUABE AND CUBE ROOTS. 11- 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


802 


643204 


515849608 


28-3196045 


9-2909072 


•001246883 


803 


644809 


517781627 


28-3372546 


9-2947671 


•001245330 


804 


646416 


619718464 


28-3548938 


9-2986239 


•001243781 


805 


648025 


521660125 


28-3725219 


9-3024775 


-001242230 


806 


649636 


523606616 


28-3901391 


9-3063278 


-001240695 


807 


651249 


625557943 


28-4077464 


9-3101750 


•001239157 


808 


652864 


627514112 


28-4253408 


9-3140190 


•001237624 


809 


654481 


629475129 


28-4429253 


9-3178699 


•001236094 


810 


656100 


631441000 


28-4604989 


9-3216975 


•001234568 


811 


657721 


633411731 


28-4780617 


9-3255320 


•001233046 


812 


659344 


535387328 


28-4956137 


9-3293634 


•001231527 


813 


660969 


637367797 


28-5131549 


9-3331916 


•001230012 


814 


662596 


639353144 


28-6306852 


9-3370167 


•001228501 


815 


664225 


641343375 


28-5482048 


9-3408386 


•001226994 


816 


665856 


643338496 


28-5657137 


9-3446575 


•001225499 


817 


667489 


645338513 


28-6832119 


9-3484731 


•001223990 


818 


669124 


647343432 


28-6006993 


9-3522857 


•001222494 


819 


670761 


649353259 


28-6181760 


9-3560952 


•001221001 


820 


672400 


551368000 


28-6356421 


9-3599016 


•001219512 


821 


674041 


553387661 


28-6530976 


9-3637049 


•001218027 


822 


675684 


555412248 


28-6705424 


9-3675051 


•001216545 


823 


677329 


557441767 


28-6879716 


9-3713022 


•001215007 


824 


678976 


659476224 


28-7054002 


9-3750963 


•001213592 


825 


680625 


661515625 


28-7228132 


9-3788873 


■001212121 


826 


682276 


563559976 


28-7402157 


9-3826752 


•001210654 


827 


683929 


566609283 


28-7576077 


9-3864600 


•001209190 


828 


685584 


667663552 


28-7749891 


9-3902419 


•001207729 


829 


687241 


669722789 


28-7923601 


9-3940206 


•001206273 


830 


688900 


571787000 


28-8097206 


9-3977964 


•001204819 


831 


690561 


673856191 


28-8270706 


9-4016691 


•001203309 


832 


692224 


675930368 


28-8444102 


9-4053387 


•001201923 


833 


693889 


578009537 


28-8617394 


9-4091054 


•001200480 


834 


695556 


680093704 


28-8790582 


9-4128690 


•001199041 


835 


697225 


582182875 


28-8963666 


9-4166297 


-001197605 


836 


698896 


684277056 


28-9136646 


9-4203873 


•001196172 


837 


700569 


586376253 


28-9309523 


9-4241420 


•001194743 


838 


702244 


588480472 


28-9482297 


9-4278936 


•001193317 


839 


703921 


590589719 


28-9654967 


9-4316423 


•001191896 


840 


705600 


592704000 


28-9827535 


9-4353800 


•001190476 


841 


707281 


694823321 


29-0000000 


9-4391307 


•001189061 


842 


708964 


596947688 


29-0172363 


9-4428704 


•001187648 


843 


710649 


699077107 


29-0344623 


9-4466072 


•001186240 


844 


712336 


601211584 


29-0516781 


9-4503410 


•001184834 


845 


714025 


603351125 


29-0688837 


9-4540719 


•001183432 


846 


715716 


605495736 


29-0860791 


9-4577999 


•001182033 


847 


717409 


607645423 


29-1032644 


9-4615249 


•001180638 


848 


719104 


609800192 


29-1204396 


9-4652470 


•001179246 


849 


720801 


611960049 


29-1376046 


9-4689661 


•001177856 


850 


722500 


614125000 


29-1547595 


9-4726824 


•001176471 


851 


724201 


616295051 


29-1719043 


9-4763957 


•001175088 


852 


725904 


618470208 


29-1890390 


9-4801061 


•001173709 


853 


727609 


620650477 


29-2061637 


9-4838136 


•001172333 


854 


729316 


622835864 


29-2232784 


9-4875182 


•001170960 


855 


731025 


625026375 


29-2403830 


9-4912200 


•001169591 


856 


732736 


627222016 


29-2574777 


9-4949188 


•001108224 


857 


734449 


629422793 


29-2745623 


9-4986147 


-001166861 


858 


736164 


631628712 


29-2916370 


9-5023078 


-001105501 


859 


737881 


633839779 


29-3087018 


9-5059980 


•001164144 


860 


739600 


636056000 


29-3257666 


9-5096854 


•001162791 


861 


741321 


638277381 


29-3428015 


9-5133699 


•001161440 


862 


743044 


640603928 


29-3698365 


9-6170515 


•001160003 


863 


744769 


642735647 


29-3768616 


9-6207303 


•001158749 



k2 



114 



THE PRACTICAL MODEL CALCULATOR. 



Numl:)er. 


Squares. 


Cul)es. 


Sq-aare Roots. 


Cube Koots. 


Reciprocals. 


864 


746496 


644972544 


29-3938769 


9-5244063 


•001157407 


865 


748225 


647214625 


29-4108823 


9-5280794 


•001156069 


866 


749956 


6494G1896 


29*4278779 


9-5317497 


•001154734 


867 


751689 


651714363 


29-4448637 


9-5354172 


•001153403 


868 


753424 


653972032 


29-4618397 


9-5390818 


•001152074 


869 


755161 


656234909 


29-4788059 


9-5427437 


•001150748 


870 


766900 


658503000 


29-4957624 


9-5464027 


•001149425 


871 


758641 


6G0776311 


29-5127091 


9-5500589 


•001148106 


872 


760384 


663054848 


29-5296461 


9-5537123 


•001146789 


873 


762129 


665338617 


29-5465734 


9-5573630 


•001145475 


874 


768876 


667627624 


29-5634910 


9-5610108 


•001144165 


875 


765625 


669921875 


29-5803989 


9-5646559 


•001142857 


.876 


767376 


672221376 


29-5972972 


9-5682782 


•001141553 


.877 


769129 


674526133 


29-6141858 


9-5719377 


•001140251 


878 


770884 


676836152 


29-6310648 


9-5755745 


•001138952 


879 


772641 


679151439 


29-6479342 


9-5792085 


•001137656 


880 


774400 


681472000 


29-6647939 


9-5828397 


•001186364 


881 


776161 


683797841 


29-6816442 


9-5864682 


•001135074 


882 


777924 


686128968 


29-6984848 


9-5900987 


•001138787 


883 


779689 


688465387 


29-7153159 


9-5987169 


•001132503 


884 


781456 


690807104 


29-7321375 


9-5973373 


•001131222 


885 


783225 


693154125 


29-7489496 


9-6009548 


•001129944 


886 


784996 


695506456 


29-7657521 


9-6045696 


•001128668 


887 


786769 


697864103 


29-7825452 


9-6081817 


•001127396 


888 


788544 


700227072 


29-7993289 


9-6117911 


•001126126 


889 


790321 


702595369 


29-8161030 


9-6153977 


•001124859 


890 


792100 


704969000 


29-8328678 


9-6190017 


•001128596 


891 


793881 


707347971 


29-8496231 


9-6226030 


•001122334 


892 


795664 


707932288 


29-8663690 


9-6262016 


•001121070 


893 


797449 


712121957 


29-8831056 


9-6297975 


•001119821 


894 


799236 


714516984 


29-8998328 


9-6833907 


•001118568 


895 


801025 


716917375 


29-9165506 


9-6369812 


•001117818 


896 


802816 


719323136 


29-9332591 


9-6405690 


•001116071 


897 


804609 


721734273 


29-9499583 


9-6441542 


•001114827 


898 


806404 


724150792 


29-9666481 


9-6477367 


•001113586 


899 


808201 


726572699 


29-9833287 


9-6513166 


•001112347 


900 


810000 


729000000 


30-0000000 


9-6548938 


•001111111 


901 


811801 


731432701 


30-0166621 


9-6584684 


•001109878 


902 


813604 


733870808 


30-0333148 


9-6620403 


•001108647 


903 


815409 


736314327 


30-0499584 


9-6656096 


•001107420 


904 


817216 


738763264 


30-0665928 


9-6691762 


•001106195 


905 


819025 


741217625 


30-0832179 


9-6727403 


•001104972 


906 


820836 


743677416 


30-0998339 


9-6763017 


•001103753 


907 


822649 


746142643 


30-1164407 


9-6798604 


•001102536 


908 


824464 


748613312 


30-1330383 


9-6834166 


•001101322 


909 


826281 


751089429 


30-1496269 


9-6869701 


•001100110 


910 


828100 


753571000 


30-1662063 


9-6905211 


•001098901 


911 


829921 


756058031 


30-1827765 


9-6940694 


•001097695 


912 


831744 


758550825 


30-1993377 


9-6976151 


•001096491 


913 


833569 


761048497 


30-2158899 


9-7011583 


•001095290 


914 


835396 


763551944 


30-2324329 


9-7046989 


•001094092 


915 


837225 


766060875 


30-2489669 


9-7082869 


•001092896 


916 


839056 


768575296 


30-2654919 


9-7117723 


•001091703 


917 


840889 


771095213 


30-2820079 


9-7153051 


•001090513 


918 


842724 


773620632 


30-2985148 


9-7188354 


•001089325 


919 


844561 


776151559 


30-3150128 


9-7228631 


■001088139 


920 


846400 


778688000 


30-3315018 


9-7258883 


•001086957 


921 


848241 


781229961 


80-3479818 


9-7294109 


•001085776 


922 


850084 


783777448 


30-3644529 


9-7329309 


•001084599 


923 


851929 


786330467 


30-3809151 


9-7364484 


•001083423 


924 


853776 


788889024 


80-3973683 


9-7399634 


•001082251 


925 


855625 


791453125 


80-4138127 


9-7434758 


•001081081 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS 



iNni'ber. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


926 


857476 


794022776 


30-4302481 


9-7469857 


•001079914 


927 


859329 


796597983 


30-4466747 


9-7504930 


•001078749 


928 


861184 


799178752 


30-4630924 


9-7539979 


•001077586 


929 


863041 


801765089 


30-4795013 


9-7575002 


•001076426 


930 


864900 


804357000 


30-4959014 


9-7610001 


-001075269 


931 


866761 


806954491 


30-5122926 


9-7644974 


•001074114 


932 


868624 


809557568 


30-5286750 


9-7679922 


•001072961 


933 


870489 


812166237 


30-5450487 


9-7714845 


•001071811 


934 


«72356 


814780504 


30-5614136 


9-7749743 


•001070664 


935 


874225 


817400375 


30-5777697 


9-7784616 


•001069519 


936 


876096 


820025856 


30-5941171 


9-7829466 


•001068376 


937 


877969 


822656953 


30-6104557 


9-7854288 


•001067236 


938 


879844 


825293672 


30-6267857 


9-7889087 


•001066098 


939 


881721 


827936019 


30-6431069 


9-7923861 


•001064963 


940 


883600 


830584000 


30-6594194 


9-7958611 


•001063830 


941 


885481 


833237621 


30-6757233 


9-7993336 


•001062699 


942 


887364 


835896888 


30-6920185 


9-8028036 


•001061571 


943 


889249 


838561807 


30-7083051 


9-8062711 


•001060445 


944 


891136 


841232384 


30-7245830 


9-8097362 


•001059322 


945 


893025 


843908625 


30-7408523 


9-8131989 


•001058201 


946 


894916 


846590536 


80-7571130 


9-8166591 


•001057082 


947 


896808 


849278123 


30-7733651 


9-8201169 


•001055966 


948 


898704 


851971392 


30-7896086 


9-8235723 


•001054852 


949 


900601 


854670349 


30-8058436 


9-8270252 


•001053741 


950 


902500 


857375000 


30-8220700 


9-8304757 


•001052632 


951 


904401 


860085351 


30-8382879 


9-8339238 


•001051525 


952 


906304 


862801408 


30-8544972 


9-8373695 


•001050420 


953 


908209 


865523177 


30-8706981 


9-8408127 


•001049318 


954 


910116 


868250664 


30-8868904 


9-8442536 


•001048218 


955 


912025 


870983875 


80-9030743 


9-8476920 


•001047120 


956 


913936 


873722816 


30-9192477 


9-8511280 


•001046025 


957 


915849 


876467493 


30-9354166 


9-8545617 


•001044932 


958 


917764 


879217912 


30-9515751 


9-8579929 


•001043841 


959 


919681 


881974079 


30-9677251 


9-8614218 


•001042753 


960 


921600 


884736000 


80-9838668 


9-8648483 


•001041667 


961 


923521 


887503681 


31-0000000 


9-8682724 


•001040583 


962 


925444 


890277128 


31-0161248 


9-8710941 


•001039501 


963 


927369 


893056347 


31-0322413 


9-8751135 


•001038422 


964 


929296 


895841344 


31-0483494 


9-8785305 


-001037344 


965 


931225 


898632125 


31-0644491 


9-8819451 


-001036269 


960 


933156 


901428696 


31-0805405 


9-8853574 


•001035197 


967 


935089 


904231063 


31-0966236 


9-8887673 


•001034126 


968 


937024 


907039232 


31-1126984 


9-8921749 


•001033058 


969 


938961 


909853209 


31-1287648 


9-8955801 


•001031992 


970 


940900 


912673000 


81-1448230 


9-8989830 


•001030928 


971 


942841 


915498611 


81-1608729 


9-9023835 


•001029806 


972 


944784 


918330048 


31-1769145 


9-9057817 


•001028807 


973 


946729 


921167317 


31-1929479 


9-9091776 


•001027749 


974 


948676 


924010424 


31-2089731 


9-9125712 


•001020694 


975 


950625 


926859375 


31-2249900 


9-9159624 


•001025641 


976 


952576 


929714176 


81-2409987 


9-9193513 


•001024590 


977 


954529 


932574833 


31-2569992 


9-9227379 


•001023541 


978 


956484 


935441352 


31-2729915 


9-9261222 


•001022495 


979 


958441 


938313739 


31-2889757 


9-9295042 


•001021450 


980 


960400 


941192000 


31-3049517 


9-9328839 


-001020408 


981 


962361 


944076141 


31-3209195 


9-9362613 


•001019168 


982 


964324 


946906168 


31-3368792 


9-9890363 


-001018330 


983 


960289 


949862087 


31-3528308 


9-9430092 


•001017294 


984 


968256 


952763904 


31-3687743 


9-9403797 


•001010260 


985 


970225 


955671625 


81-3847097 


9-9497479 


•001015228 


986 


972196 


958585256 


31-4006369 


9-9531138 


•001014199 


987 


974169 


961504803 


31-4165561 


9-9504775 


•001013171 



116 



THE PKACTICAL MODEL CALCULATOR, 



Number. 


Squares. 


Culjes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


988 


976144 


964430272 


31-4324673 


9-9598389 


•001012146 


989 


978121 


967361669 


31-4483704 


9-9631981 


-001011122 


990 


980100 


970299000 


31-4642654 


9-9665549 


•001010101 


991 


982081 


973242271 


31-4801525 


9-9699055 


-001009082 


992 


984064 


976191488 


31-4960315 


9-9732619 


-001008065 


993 


986049 


979146657 


31-5119025 


9-9766120 


•001007049 


994 


988036 


982107784 


31-5277655 


9-9799599 


•001006036 


995 


990025 


985074875 


31-5436206 


9-9833055 


•001005025 


996 


992016 


988047936 


31-5594677 


9-9866488 


•001004016 


997 


994009 


991026973 


31-5753068 


9-9899900 


•001003009 


998 


996004 


994011992 


31-5911380 


9-9933289 


•001002004 


999 


998001 


997002999 


31-6069613 


9-9966656 


•001001001 


1000 


1000000 


1000000000 


31-6227766 


10-0000000 


•001000000 


1001 


1000201 


1003003001 


31-6385840 


10-0033222 


•0009990010 


1002 


1004004 


1006012008 


81-6543866 


10-0066622 


•0009980040 


1003 


1006009 


1009027027 


31-6701752 


10-0099899 


•0009970090 


1004 


1008016 


1012048064 


31-6859590 


10-0133155 


•0009960159 


1005 


1010025 


1015075125 


31-7017349 


10-0166389 


•0009950249 


1006 


1010036 


1018108216 


31-7175080 


10-0199601 


•0009940358 


1007 


1014049 


1021147343 


31-7832633 


10-0232791 


•0009930487 


1008 


1016064 


1024192512 


31-7490157 


10-0265958 


•0009920685 


1009 


1018081 


1027243729 


31-7647603 


10-0299104 


•0009910803 


1010 


1020100 


1030301000 


31-7804972 


10-0332228 


•0009900990 


1011 


1020121 


1033364331 


31-7962262 


10-0365330 


•0009891197 


1012 


1024144 


1036433728 


31-8119474 


10-0398410 


•0009881423 


1013 


1026169 


1039509197 


31-8276609 


10-0431469 


•0009871668 


1014 


1028196 


1042590744 


31-8433666 


10-0464506 


•0009861933 


1015 


1030225 


1045678375 


31-8590646 


10-0497521 


•0009852217 


1016 


1032256 


1048772096 


31-8747549 


10-0530514 


■0009842520 


1017 


1034289 


1051871913 


31-8904374 


10-0563485 


•0009832842 


1018 


1036324 


1054977832 


31-9061123 


10-0596435 


•0009823183 


1019 


1038361 


1058089859 


31-9217794 


10-0629364 


•0009813543 


1020 


1040400 


1061208000 


31-9374388 


10-0662271 


-0009803922 


1021 


1042441 


1064332261 


31-9530906 


10-0695156 


-0009794319 


1022 


1044484 


1067462648 


31-9687347 


10-0728020 


•0009784736 


1028 


1046529 


1070599167 


31-9843712 


10-0760863 


•0009775171 


1024 


1048576 


1073741824 


32-0000000 


10-0793684 


•0009765625 


1025 


1050625 


1076890625 


32-0156212 


10-0826484 


•0009756098 


1026 


1052676 


1080045576 


82-0312348 


10-0859262 


•0009746589 


1027 


1054729 


1083206683 


32-0468407 


10-0892019 


•0009737098 


1028 


1056784 


1086373952 


32-0624391 


100924755 


•0009727026 


1029 


1058841 


1089547389 


32-0780298 


10-0957469 


•0009718173 


1030 


1060900 


1092727000 


32-0936131 


10-0990163 


-0009708738 


1031 


1062961 


1095912791 


82-1091887 


10-1022835 


•0009699321 


1032 


1065024 


1099104768 


82-1247568 


10-1055487 


•0009689922 


1033 


1067089 


1102302937 


82-1403173 


10-1088117 


•0009680542 


1034 


1069156 


1105507304 


32-1558704 


10-1120726 


•0009671180 


1035 


1071225 


1108717875 


82-1714159 


10-1153314 


•0009661886 


1036 


1073296 


1111934656 


32-1869539 


10-1185882 


•0009652510 


1037 


1075369 


1115157653 


32-2024844 


10-1218428 


•0009643202 


1038 


1077444 


1118386872 


32-2] 80074 


10-1250953 


-0009633911 


1039 


1079521 


1121622319 


82-2335229 


10-1283457 


-0009624639 


1040 


1081600 


1124864000 


82-2490310 


10-1315941 


-0009615385 


1041 


1083681 


1128111921 


32-2645316 


10-1348403 


-0009606148 


1042 


1085764 


1131366088 


32-2800248 


10-1880845 


-0009596929 


1043 


1087849 


1134626507 


32-2955105 


10-1413266 


-0009587738 


1044 


1039936 


1137893184 


82-3109888 


10-1445667 


-0009578544 


1045 


1092025 


1141166125 


32-3264598 


10-1478047 


-0009569378 


1046 


1094116 


1144445336 


32-3419233 


10-1510406 


•0009560229 


1047 


1096209 


1147730823 


32-3573794 


10-1542744 


•0009551098 


1048 


1098304 


1151022592 


82-3728281 


10-1575082 


-0009541985 


1049 

1 


1100401 


1154320649 


32-3882695 


10-1607359 


•0009532888 



TABLE GF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 117 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1050 


1102500 


1157625000 


32-4037035 


10-1639636 


•0009523810 


1051 


1104001 


1160935651 


32-4191301 


10-1671893 


•0009514748 


1052 


1106704 


1164252608 


32-4345495 


10-1704129 


•0009505703 


1053 


1108809 


1167575877 


32-4499615 


10-1736344 


•0009496676 


1054 


1110916 


1170905464 


32-4653662 


10-1768539 


•0009487666 


1055 


1113125 


1174241375 


32-4807635 


10-1800714 


•0009478673 


1056 


1115130 


1177583616 


32-4961536 


10-1832868 


•0009469697 


1057 


1117249 


1180932193 


32-5115364 


10-1865002 


•0009460738 


1058 


1119364 


1184287112 


32-5269119 


10-1897116 


-0009451796 


1059 


1121481 


1187648379 


32-5422802 


10-1929209 


•0009442871 


1060 


1123600 


1191016000 


32-5576412 


10-1961283 


•0009433962 


1061 


1125721 


1194389981 


32-5729949 


10-1993336 


•0009425071 


1062 


1127844 


1197770328 


32-5883415 


10-2025369 


•0009416196 


1063 


1129969 


1201157047 


32-6035807 


10-2057382 


•0009407338 


1064 


1132096 


1204550144 


32-6190129 


10-2089375 


•0009398490 


1065 


1134225 


1207949625 


32-6343377 


10-2121347 


•0009389671 


1066 


1136356 


1211355496 


32-6496554 


10-2153300 


•0009380863 


1067 


1138489 


1214767763 


32-6649659 


10-2185233 


•0009372071 


1068 


1140624 


1218186432 


32-6802693 


10-2217146 


•0009363296 


1069 


1142761 


1221611509 


32-6955654 


10-2249039 


•0009354537 


1070 


1144900 


1225043000 


32-7108544 


10-2280912 


•0009345794 


1071 


1147041 


1228480911 


32-7261363 


10-2312766 


•0009337068 


1072 


1149184 


1231925248 


32-7414111 


10-2344599 


•0009328358 


1073 


1151329 


1235376017 


32-7566787 


10-2376413 


•0009319664 


1074 


1153476 


1238833224 


32-7719392 


10-2408207 


•0009310987 


1075 


1155625 


1242296875 


32-7871926 


10-2439981 


•0009302326 


1076 


1157776 


1245766976 


32-8024398 


10-2471735 


•0009293680 


1077 


1159929 


1249243533 


32-8176782 


10-2503470 


•0009285051 


1078 


1162084 


1252726552 


32-8329103 


10-2535186 


•0009276438 


1079 


1164241 


1256216039 


32-8481354 


10-2566881 


•0009267841 


1080 


1166400 


1259712000 


32-8633535 


10-2598557 


•0009259259 


1081 


1168561 


1263214441 


32-8785644 


10-2630213 


•0009250694 


1082 


1170724 


1266723368 


32-8937684 


10-2661850 


•0009242144 


1083 


1172889 


1270238787 


32-9089653 


10-2693467 


•0009233610 


1084 


1175056 


1273760704 


32-9241553 


10-2725065 


•0009225092 


1085 


1177225 


1277289125 


32-9393382 


10-2756644 


•0009216590 


1086 


1179396 


1280824056 


32-9545141 


10-2788203 


•0009208103 


1087 


1181569 


1284365503 


32-9696830 


10-2819743 


•0009199632 


1088 


1183744 


1287913472 


32-9848450 


10-2851264 


•0009191176 


1089 


1185921 


1291467969 


33-0000000 


10-2882765 


•0009182736 


1090 


1188100 


1295029000 


33-0151480 


10-2914247 


•0009174312 


1091 


1190281 


1298596571 


33-0302891 


10-2945709 


•0009165903 


1092 


1192464 


1302170688 


33-0454233 


19-2977153 


•0009157509 


1093 


1194649 


1305751357 


33-0605505 


10-3008577 


•0009149131 


1094 


1196836 


1309338684 


33-0756708 


10-3039982 


•0009140768 


1095 


1199025 


1312932375 


33-0907842 


10-3071368 


•0009132420 


1096 


1201216 


1316532736 


33-1058907 


10-3102735 


•0009124008 


1097 


1203409 


1320139673 


33-1209903 


10-3134083 


•0009115770 


1098 


1205604 


1323753192 


33-1300830 


10-3165411 


•0009107468 


1099 


1207801 


1327373299 


33-1511689 


10-3190721 


•0009099181 


1100 


1210000 


1331000000 


33-1662479 


10-3228012 


•0009090909 


1101 


1212201 


1334633301 


33-1813200 


10-3259284 


•0009082652 


1102 


1214404 


1338273208 


33-1963853 


10-3290537 


•0009074410 


1103 


1216609 


1341919727 


33-2114438 


10-3321770 


•0009066183 


1104 


1218816 


1345572864 


33-2266955 


10-3352985 


•0009057971 


1105 


1221025 


1349232625 


33-2415403 


10-3384181 


•0009049774 


1106 


1223236 


1352899016 


33-2565783 


10-3415358 


•0009041591 


1107 


1225449 


135G572043 


33-2716095 


10-3446517 


•0009033424 


1108 


1227664 


1360251712 


33-2806339 


10-3477657 


-0009025271 


1109 


1229881 


1363938029 


33-3016516 


10-3508778 


-0009017133 


1110 


1232100 


1367631000 


33-3166625 


10-3539880 


-0009009009 


nil 


1234321 


1371330631 


33-3316666 


10-3570964 


•0009000900 



118 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


Square Eoots. 


C-abe Roots. 


Reciprocals. 


1112 


1236544 


1375036928 


33-3466640 


10-3602029 


-0008992806 


1113 


1238769 


1378749897 


33-3616546 


10-3633076 


•0008984726 


1114 


1240996 


1382469544 


33-3766385 


10-3664103 


-0008976661 


1115 


1243225 


1386195875 


33-3916157 


10-3695113 


•0008968610 


1116 


1245456 


1389928896 


33-4065862 


10-3726103 


-0008960753 


1117 


1247689 


1393668613 


33-4215499 


10-3757076 


•0008952551 


1118 


1249924 


1397415032 


33-4365070 


10-3788030 


•0008944544 


1119 


1252161 


1401168159 


33-4514573 


10-3818965 


•0008936550 


1120 


1254400 


1404928000 


33-4664011 


10-3849882 


•0008928571 


1121 


1256641 


1408694561 


33-4813381 


10-3880781 


•0008960607 


1122 


1258884 


1412467848 


33-4962684 


10-3911661 


•0008912656 


1123 


1261129 


1416247867 


33-5111921 


10-3942527 


-0008904720 


1124 


1263376 


1420034624 


33-5261092 


10-3973366 


-0008896797 


1125 


1265625 


1423828125 


33-5410196 


10-4004192 


•0008888889 


1126 


1267876 


1427628376 


33-5559234 


10-4034999 


•0008880995 


1127 


1270129 


1431435383 


33-5708206 


10-4065787 


•0008873114 


1128 


1272384 


1435249152 


33-5857112 


10-4096557 


•0008865248 


1129 


1274641 


1439069689 


33-6005952 


10-4127310 


•0008857396 


1130 


1276900 


1442897000 


33-6154726 


10-4158044 


•0008849558 


1131 


1279161 


1446731091 


33-6303434 


10-4188760 


•0008841733 


1132 


1281424 


1450571968 


33-6452077 


10-4219458 


•0008833922 


1133 


1283689 


1454419637 


33-6600653 


10-4250138 


•0008826125 


1134 


1285956 


1458274104 


33-6749165 


10-4280800 


-0008818342 


1135 


1288225 


1462135375 


33-6897610 


10-4311443 


-0008810573 


1136 


1290496 


1466003456 


33-7045991 


10-4342069 


-0008802817 


1137 


1292769 


1469878353 


33-7174306 


10-4372677 


-0008795075 


1138 


1295044 


1473760072 


33-7340556 


10-4403677 


•0008787346 


1139 


1297321 


1477648619 


33-7490741 


10-4433839 


•0008779631 


1140 


1299600 


1481544000 


33-7638860 


10-4464393 


•0008771930 


1141 


1301881 


1485446221 


33-7786915 


10-4494929 


-0008764242 


1142 


1304164 


1489355288 


33-7934905 


10-4525448 


-0008756567 


1143 


1306449 


1493271207 


33-8082830 


10-4555948 


-0008748906 


1144 


1308736 


1497193984 


33-8230691 


10-4586431 


•0008741259 


1145 


1311025 


1501123625 


33-8378486 


10-4616896 


•0008733624 


1146 


1313316 


1505060136 


33-8526218 


10-4647343 


•0008726003 


1147 


1315609 


1509003523 


33-8673884 


10-4677773 


•0008718396 


1148 


1317904 


1512953792 


33-8821487 


10-4708158 


-0008710801 


1149 


1320201 


1516910949 


33-8969025 


10-4738579 


-0008703220 


1150 


1322500 


1520875000 


33-9116499 


10-4768955 


•0008695652 


1151 


1324801 


1524845951 


33-9263909 


10-4799314 


•0008688097 


1152 


1327104 


1528823808 


33-9411255 


10-4829656 


•0008680556 


1153 


1329409 


1532808577 


33-9558537 


10-4859980 


-0008673027 


1154 


1331716 


1536800264 


33-9705755 


10-4890286 


•0008665511 


1155 


1334025 


1540798875 


33-9852910 


10-4920575 


•0008658009 


1156 


1336336 


1544804416 


34-0000000 


10-4950847 


•0008650519 


1157 


1338649 


1548816893 


34-0147027 


10-4981101 


•0008643042 


1158 


1340964 


1552836312 


34-0293990 


10-5011337 


•0008635579 


1159 


1343281 


1556862679 


34-0440890 


10-5041556 


•0008628128 


1160 


1345600 


1560896000 


84-0587727 


10-5071757 


•0008620690 


1161 


1347921 


1564936281 


34-0734501 


10-5101942 


•0008613244 


1162 


1350244 


1568983528 


34-0881211 


10-5132109 


•0008605852 


1163 


1352569 


1573037749 


34-0127858 


10-5162259 


•0008598452 


1164 


1354896 


1577098944 


34-1174442 


10-5192391 


•0008591065 


1165 


1357225 


1581167125 


34-1320963 


10-5222506 


•0008583691 


1166 


1359556 


1585242296 


34-1467422 


10-5252604 


-0008576329 


1167 


1361889 


1589324463 


34-1613817 


10-5282685 


•0008568980 


1168 


1364224 


1593413632 


34-1760150 


10-5312749 


•0008561644 


1169 


1366561 


1597509809 


34-1906420 


10-5342795 


•0008554320 


1170 


1368900 


1001613000 


34-2052627 


10-5372825 


•0008547009 


1171 


1371241 


1605723211 


34-2198773 


10-5402837 


•0008539710 


1172 


1373584 


1609810418 


34-2344855 


10-5432832 


•0008532423 


1173 


1375929 


1613964717 


34-2490875 


10-5462810 


•0008525149 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 119 



Number. | 


Squares, j 


Cubes. 1 


Square Roots. 


Cube Roots. 


Reciprocals. 


1171 


1378276 


1618096024 


34-2636884 


10-549277> 


-0008517888 


1175 


1380625 


1622234375 


34-2782730 


10-5522715 


-0008510G38 


1176 


1382976 


1626379776 


34-2928564 


10-5552642 


•0008503401 


1177 


1385329 


1630532233 


84-3074336 


10-5582552 


-0008496177 


1178 


1387684 


1634691752 


84-3220046 


10-5612445 


-0008488904 


1179 


1390041 


1638858339 


84-3365694 


10-5642322 


•0008481764 


1180 


1392400 


1643032000 


34-3511281 


10-5672181 


-0008471576 


1181 


1394761 


1647212741 


84-8656805 


10-5702024 


•0008467401 


1182 


1397124 


1651400568 


34-3802268 


10-5731849 


-0008460237 


1183 


1399489 


1655595487 


84-3947670 


10-5761658 


•0008453085 


1184 


1401856 


1659797504 


84-4093011 


10-5791449 


-0008445946 


1185 


1404225 


1664006625 


34-4288289 


10-5821225 


•0008438819 


1186 


1406596 


1668222856 


34-4383507 


10-5850983 


•0008431708 


1187 


1408969 


1672446203 


34-4528668 


10-5880725 


•0008424600 


1188 


1411344 


1676676672 


84-4678759 


10-5910450 


•0008417508 


1189 


1413721 


1680914629 


34-4818793 


10-5940158 


-0008410429 


1190 


1416100 


1685159000 


34-4963766 


10-5969850 


•0008403361 


1191 


1418481 


1689410871 


34-5108678 


10-5999525 


•0008396306 


1192 


1420864 


1693669888 


84-5258530 


10-6029184 


-0008389262 


1193 


1423249 


1697936057 


34-5398321 


10-6058826 


•0008382320 


1194 


1425636 


1702209384 


34-5543051 


10-6088451 


•0008375209 


1195 


1428025 


1706489875 


34-5687720 


10-6118060 


•0008368201 


1196 


1430416 


1710777536 


34-5832329 


10-6147652 


•0008361204 


1197 


1432809 


1715072373 


34-5976879 


10-6177228 


•0008354219 


1198 


1435204 


1719374392 


84-6121366 


10-6206788 


-0008347245 


1199 


1437601 


1723683599 


34-6265794 


10-6236331 


•0008340284 


1200 


1440000 


1728000000 


34-6410162 


10-6265857 


•0008333333 


1201 


1442401 


1732323601 


84-6554469 


10-6295367 


•0008326395 


1202 


1444804 


1736654408 


34-6698716 


10-6824860 


•0008319468 


1203 


1447209 


1740992427 


34-6842904 


10-6354338 


•0008312552 


1204 


1449016 


1745337664 


34-6987031 


10-6383799 


•0008305648 


1205 


1452025 


1749690125 


84-7131099 


10-6413244 


•0008298755 


1206 


1454436 


1754049816 


34-7275107 


10-6442672 


•0008291874 


1207 


1456849 


1758416743 


84-7419055 


10-6472085 


•0008285004 


1208 


1459264 


1762790912 


34-7562944 


10-6501480 


•0008278146 


1209 


1461681 


1767172329 


34-7706773 


10-6530860 


•0008271299 


1210 


1464100 


1771561000 


34-7850543 


10-6560223 


•0008264463 


1211 


1466521 


1775956931 


34-7994258 


10-6589570 


•0008257638 


1212 


1468944 


1780360128 


34-8137904 


10-6618902 


•0008250825 


1213 


1471369 


1784770597 


34-8281495 


10-6648217 


•0008244023 


1214 


1473796 


1789188344 


84-8425028 


10-6677516 


•0008237232 


1215 


1476225 


1793613375 


34-8568501 


10-6706799 


•0008230453 


1216 


1478656 


1798045696 


34-8711915 


10-6736066 


•0008223684 


1217 


1481089 


1802485313 


34-8855271 


10-6765317 


•0008216927 


1218 


1483524 


1806932232 


34-8998567 


10-6794552 


•0008210181 


1219 


1485961 


1811386459 


34-9141805 


10-0828771 


•0008203445 


1220 


1488400 


1815848000 


34-9284984 


10-6852978 


•0008196721 


1221 


1490841 


1820316861 


34-9428104 


10-6882160 


•0008190008 


1222 


1493284 


1824793048 


84-9571166 


10-6911331 


•0008183306 


1223 


1495729 


1829276567 


84-9714109 


10-6940486 


•0008176615 


1224 


1498176 


1833764247 


34-9857114 


10-6969625 


•0008169935 


1225 


1500625 


1838265625 


35-0000000 


10-6998748 


•0008163265 


1226 


1503276 


1842771176 


85-0142828 


10-7027855 


•0008156607 


1227 


1505529 


1847284083 


85-0285598 


10-7056947 


•0008149959 


1228 


1507984 


1851804352 


85 0428309 


10-7086023 


•0008143322 


1229 


1510441 


1856331989 


35-0570963 


10-7115083 


•0008136696 


1230 


1512900 


1860867000 


35-0713558 


10-7144127 


-0008130031 


1231 


1515361 


1865409391 


85-0856096 


10-7173155 


•0008123477 


1232 


1517824 


1869959168 


85-0998575 


10-7202168 


•0008116883 


1233 


1520289 


1874516337 


85-1140997 


10-7231165 


•OOOS110300 


1234 


1522756 


1879080904 


85-1283361 


10-7260146 


•0008103728 


1235 


1525225 


1883652875 


35-1425568 


10-7289112 


•0008097166 



120 



THE PRACTICAL MODEL CALCULATOE. 



Number. 


Sqiiares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


123G 


1527696 


1888282256 


35-1567917 


10-7318062 


-0008090615 


1237 


1530169 


1892819053 


35-1710108 


10-7846997 


•0008084074 


1238 


1532644 


1897413272 


35-1852242 


10-7375916 


•0008077544 


1239 


1635121 


1902014919 


35-1994318 


10-7404819 


•0008071025 


1240 


1537600 


1906624000 


35-2136337 


10-7433707 


•0008064516 


1241 


1540081 


1911240521 


35-2278299 


10-7462579 


•0008058018 


1242 


1542564 


1915864488 


35-2420204 


10-7491436 


•0008051530 


1243 


1645049 


1920495907 


35-2562051 


10-7520277 


•0008045052 


1244 


1547536 


1925134784 


35-2703842 


10-7549103 


•0008038585 


1245 


1550025 


1929781125 


35-2845575 


10-7577913 


•0008032129 


1246 


1552521 


1934434936 


35-2987252 


10-7606708 


•0008025682 


1247 


1555009 


1939096223 


35-3128872 


10-7635488 


•0008019246 


1248 


1557504 


1943764992 


85-3270435 


10-7664252 


•0008012821 


1249 


1560001 


1948441249 


35-3411941 


10-7698001 


•0008006405 


1250 


1562500 


1953125000 


35-8558391 


10-7721735 


•0008000000 


1251 


1565001 


1957816251 


35-3694784 


10-7750453 


•0007993605 


1252 


1567504 


1962515008 


85-3836120 


10-7779156 


•0007987220 


1253 


1570009 


1967221277 


85-8977400 


10-7807843 


•0007980846 


1254 


1572516 


1971935064 


35-4118624 


10-7836516 


•0007974482 


1255 


1575025 


1976656375 


35-4259792 


10-7865173 


•0007968127 


1256 


1577536 


1981385216 


35-4400903 


10-7893815 


•0007961783 


1257 


1580049 


1986121593 


35-4541958 


10-7922441 


•0007955449 


1258 


1582564 


1990865512 


35-4682957 


10-7951053 


•0007949126 


1259 


1585081 


1995616979 


35-4823900 


10-7979649 


-0007942812 


12G0 


1587600 


2000376000 


35-4964787 


10-8008230 


•0007936508 


1261 


1590121 


2005142581 


35-5105618 


10-8036797 


•0007930214 


1262 


1592644 


2009916728 


35-5246393 


10-8065848 


•0007923930 


1263 


1595166 


2014698447 


35-5387113 


10-8093884 


•0007917656 


1264 


1597696 


2019487744 


85-5527777 


10-8122404 


•0007911392 


1265 


1600225 


2024284625 


35-5668385 


10-8150909 


•0007905188 


1266 


1602756 


2029089096 


85-5808937 


10-8179400 


•0007898894 


1267 


1005289 


2033901163 


35-5949434 


10-8207876 


•0007892660 


1268 


1607824 


2038720832 


35-6089876 


10-8236336 


•0007886435 


1269 


1610361 


2043548109 


85-6230262 


10-8264782 


•0007880221 


1270 


1612900 


2048383000 


35-6370593 


10-8293213 


•0007874016 


1271 


1615441 


2053225511 


85-6510869 


10-8321629 


•0007867821 


1272 


1617984 


2058075648 


85-6651090 


10-8350030 


•0007861635 


1273 


1620529 


2062933417 


35-6791255 


10-8378416 


•0007855460 


1274 


1623076 


2067798824 


85-6981366 


10-8406788 


•0007849294 


1275 


1625625 


2072671875 


85-7071421 


10-8435144 


•0007843137 


1276 


1628176 


2077552576 


35-7211422 


10-8463485 


•0007836991 


1277 


1630729 


2082440933 


35-7351367 


10-8491812 


•0007830854 


1278 


1633284 


2087336952 


35-7491258 


10-8520125 


•0007824726 


1279 


1635841 


2092240639 


85-7631095 


10-8548422 


•0007818608 


1280 


1638400 


2097152000 


35-7770876 


10-8576704 


•0007812500 


1281 


1640961 


2102071841 


35-7910603 


10-8604972 


•0007806401 


1282 


1643524 


2106997768 


35-8050276 


10-8033225 


•0007800312 


1283 


1646089 


2111932187 


35-8189894 


10-8661454 


•0007794232 


1284 


1648656 


2116874304 


85-8329457 


10-8689687 


•0007788162 


1285 


1651225 


2121824125 


35-8468966 


10-8717897 


•0007782101 


1286 


1653796 


2126781656 


35-8608421 


10-8746091 


•0007776050 


1287 


1656369 


2131746903 


85-8747822 


10-8774271 


•0007770008 


1288 


1658944 


2136719872 


35-8887169 


10-8802436 


•0007763975 


1289 


1661521 


2141700569 


35-9026461 


10-8830587 


•0007757952 


1290 


1664100 


2146689000 


35-9165699 


10-8858723 


•0007751938 


1291 


1666681 


2151685171 


35-9304884 


10-8886845 


•0007745933 


1292 


1669264 


2156689088 


35-9444015 


10-8914952 


•0007739938 


1293 


1671849 


2161700757 


35-9583092 


10-8943044 


•0007733952 


1294 


1674436 


2166720184 


35-9722115 


10-8971123 


•0007727975 


1295 


1677025 


2171747375 


85-9861084 


10-8999186 


•0007722008 


1296 


1679616 


2176782336 


86-0000000 


10-9027235 


•0007716019 


1297 


1682209 


2181825073 


36-0138862 


10-9055269 


-0007710100 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 121 



Numter. 


Squares. 


Cubes. 


Square Eoots. 


Cube Eoots. 


Keciprocals. 


1298 


1684804 


2186875592 


36-0277671 


10-9083290 


•0007704160 


1299 


1687401 


2191933899 


36-0416426 


10-9111296 


•0007698229 


1300 


1690000 


2197000000 


36-0555128 


10-9139287 


•0007692308 


1301 


1692601 


2202073901 


36-0693776 


10-9167265 


•0007686395 


1302 


1695204 


2207155608 


36-0832371 


10-9195228 


•0007680492 


1303 


1697809 


2212245127 


36-0970913 


10-9223177 


•0007674579 


1304 


1700416 


2217342464 


36-1109402 


10-9251111 


•0007668712 


1305 


1703025 


2222447625 


36-1247837 


10-9279031 


•0007662835 


1306 


1705636 


2227560616 


36-1386220 


10-9306937 


•0007656968 


1307 


1708249 


2232681443 


36-1524550 


10-9334829 


•0007651109 


1308 


1710864 


2237810112 


36-1662826 


10-9362706 


•0007645260 


1309 


1713481 


2242946629 


36-1801050 


10-9390569 


•0007639419 


1310 


1716100 


2248091000 


36-1939221 


10-9418418 


•0007633588 


1311 


1718721 


2253243231 


36-2077340 


10-9446263 


•0007627765 


1312 


1721344 


2258403328 


36-2215406 


10-9475074 


•0007621951 


1313 


1723969 


2263571297 


36-2353419 


10-9501880 


•0007616446 


1314 


1726596 


2268747144 


36-2491379 


10-9529673 


•0007610350 


1315 


1729225 


2273930875 


36-2626287 


10-9557451 


•0007604563 


1316 


1731856 


2279122496 


36-2767143 


10-9585215 


•0007598784 


1317 


1734489 


2284322013 


36-2904246 


10-9612965 


•0007593014 


1318 


1737124 


2289529432 


36-3042697 


10-9640701 


•0007587253 


1319 


1739761 


2294744759 


36-3180396 


10-9668423 


•0007581501 


1320 


1742400 


2299968000 


36-3318042 


10-9696131 


•0007575758 


1321 


1745041 


2305199161 


36-3455637 


10-9723825 


•0007570023 


1322 


1747684 


2310438248 


36-3593179 


10-9751505 


•0007564297 


1323 


1750329 


2315685267 


36-3730670 


10-9779171 


•0007558579 


1324 


1752976 


2320940224 


36-3868108 


10-9806823 


•0007552870 


1325 


1755625 


2326203125 


36-4005494 


10-9834462 


•0007547170 


1326 


1758276 


2331473976 


36-4142829 


10-9862086 


•0007541478 


1327 


1760929 


2336752783 


36-4280112 


10-9889696 


•0007535795 


1328 


1763584 


2342039552 


36-4417343 


10-9917293 


•0007530120 


1329 


1766241 


2347334289 


36-4554523 


10-9944876 


•0007524454 


1330 


1768900 


2352637000 


36-4691650 


10-9972445 


•0007518797 


1331 


1771561 


2357947691 


36-4828727 


11-0000000 


•0007513148 


1332 


1774224 


2363266368 


36-4965752 


11-0027541 


•0007507508 


1333 


1776889 


2368593037 


36-5102725 


11-0055069 


•0007501875 


1334 


1779556 


2373927704 


36-5239647 


11-0082583 


•0007496252 


1335 


1782225 


2379270375 


36-5376518 


11-0110082 


•0007490637 


1336 


1784896 


2384621056 


36-5513388 


11-0137569 


•0007485030 


1337 


1787569 


2389979753 


36-5650106 


11-0165041 


•0007479432 


1338 


1790244 


2395346472 


36-5786823 


11-0192500 


•0007473842 


1339 


1792921 


2400721219 


36-5923489 


11-0219945 


-0007468260 


1340 


1795600 


2406104000 


36-6060104 


11-0247377 


•0007462687 


1341 


1798281 


2411494821 


36-6196668 


11-0274795 


•0007457122 


1342 


1800964 


2416893688 


36-6333181 


11-0302199 


•0007451565 


1343 


1803649 


2422300607 


30-6469144 


11-0329590 


•0007446016 


1344 


1806336 


2427715584 


36-6606056 


11-0356967 


•0007440476 


, 1345 


1809025 


2433138625 


30-6742416 


11-0384330 


•0007434944 


1346 


18*1716 


2438569736 


36-6878726 


11-0411080 


•0007429421 


1347 


1814409 


2444008923 


36-7014986 


11-0439017 


•0007423905 


1348 


1817104 


2449456192 


30-7151195 


11-0466339 


•0007418398 


1349 


1819801 


2454911549 


36-7287353 


11-0493649 


•0007412898 


1350 


1822500 


2460375000 


36-7423461 


11-0520945 


•0007407407 


1351 


1825201 


2405846551 


36-7559519 


11-0548227 


•0007401924 


1352 


1827904 


2471326208 


36-7695526 


11-0575497 


•0007396450 


1353 


1830609 


2476813977 


36-7831483 


11-0602752 


•0007390983 


1354 


1833310 


2482309864 


36-7967390 


11-0629994 


•0007385524 


1355 


1836025 


2487813875 


36-8103246 


11-0657222 


•0007380074 


1356 


1838736 


2493326016 


36-8239053 


11-0684437 


•0007374631 


1357 


1841449 


2498846293 


36-8374809 


11-0711039 


•0007369197 


1358 


1844164 


2504374712 


36-8510515 


11-0738828 


•0007363770 


1359 


1846881 


2509911279 


30-8646172 


11-0766003 


•0007358352 



122 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


IStiO 


1849600 


2515456000 


36-8781778 


11-0793165 


•0007352941 


1361 


1852321 


2521008881 


36-8917335 


11-0820314 


•0007347539 


1362 


1855044 


2526569928 


36-9052842 


11-0847449 


•0007342144 


1363 


1857769 


2532139147 


36-9188299 


11-0874571 


•0007836757 


1364 


1860496 


2537716544 


36-9323706 


11-0901679 


•0007331878 


1365 


1863225 


2543302125 


36-9459064 


11-0928775 


•0007326007 


1366 


1865956 


2548895896 


36-9594372 


11-0955857 


-0007320644 


1367 


1868689 


2554497863 


36-9729631 


11-0982926 


•0007335289 


1368 


1871424 


2560108032 


36-9864840 


11-1009982 


-0007309942 


1369 


1874161 


2565726409 


37-0000000 


11-1037025 


-0007304602 


1370 


1876900 


2571353000 


37-0135110 


11-1064054 


-0007299270 


1371 


1879641 


2576987811 


37-0270172 


11-1091070 


-0007293946 


1372 


1882384 


2582630848 


37-0405184 


11-1118073 


-0007288630 


1373 


1885129 


2588282117 


37-0540146 


11-1145064 


•0007283321 


1374 


1887876 


2593941624 


37-0675060 


11-1172041 


•0007278020 


1375 


1890625 


2599609375 


87-0899924 


11-1199004 


•0007272727 


1376 


1893376 


2605285376 


37-0944740 


11-1225955 


•0007267442 


1377 


1896129 


2610969633 


37-1079506 


11-1252893 


•0007262164 


1378 


1898884 


2616662152 


37-1214224 


11-1279817 


-0007256894 


1379 


1901641 


2622362939 


87-1348893 


11-1306729 


•0007251632 


1380 


1904400 


2628072000 


37-1483512 


11-1333628 


-0007246377 


1381 


1907161 


2633789341 


37-1618084 


11-1360514 


•0007241130 


1382 


1909924 


2639514968 


87-1752606 


11-1387386 


•0007235890 


1383 


1912689 


2645248887 


37-1887079 


11-1414246 


•0007230658 


1384 


1915456 


2650991104 


37-2021505 


11-1441093 


•0007225434 


1385 


1918225 


2656741625 


37-2155881 


11-1467926 


-0007220217 


1386 


1920996 


2662500456 


37-2290209 


11-1494747 


•0007215007 


1387 


1923769 


2668267603 


37-2424489 


11-1521555 


-0007209805 


1388 


1926544 


2674043072 


87-2558720 


11-1548350 


•0007204611 


1389 


1929321 


2679826869 


37-2692903 


11-1575133 


•0007199424 


1390 


1932100 


2685619000 


37-2827037 


11-1601903 


-0007194245 


1391 


1934881 


2691419471 


87-2961124 


11-1628659 


•0007189073 


1392 


1937664 


2697228288 


37-3095162 


11-1655403 


-0007183908 


1393 


1940449 


2703045457 


37-3229152 


11-1682134 


•0007178751 


1394 


1943236 


2708870984 


37-3363094 


11-1708852 


-0007173601 


1395 


1946025 


2714704875 


87-3496988 


11-1735558 


-0007168459 


1396 


1948816 


2720547136 


37-3630834 


11-1762250 


•0007163324 


1397 


1951609 


2726397778 


37-3764632 


11-1788930 


•0007158196 


1398 


1954404 


2732256792 


37-3898382 


11-1815598 


-0007153076 


1399 


1957201 


2738124199 


87-4032084 


11-1842252 


•0007147963 


1400 


1960000 


2744000000 


37-4165738 


11-1868894 


•0007142857 


1401 


1962801 


2749884201 


37-4299345 


11-1895523 


•0007137759 


1402 


1965604 


2755776808 


87-4432904 


11-1922139 


•0007132668 


1403 


1968409 


2761677827 


87-4566416 


11-1948743 


•0007127584 


1404 


1971216 


2767587264 


87-4699880 


11-1975334 


•0007122507 


1405 


1974025 


2773505123 


37-4833296 


11-2001913 


•0007117438 


1406 


1976836 


2779431416 


87-4966665 


11-2028479 


•0007112376 


1407 


1979649 


2785366143 


37-5099987 


11-2055032 


•0007107321 


1408 


1982464 


2791309312 


87-5233261 


11-2081573 


• -0007102273 


1409 


1985281 


2797260929 


37-5366487 


11-2108101 


•0007097282 


1410 


1988100 


2803221000 


37-5499667 


11-2134617 


•0007092199 


1411 


1990921 


2809189531 


87-5632799 


11-2161120 


-0007087172 


1412 


1993744 


2815166528 


87-5765885 


11-2187611 


-0007082158 


1413 


1996569 


2821151997 


37-5898922 


11-2214089 


-0007077141 


1414 


1999396 


2827145944 


37-6031913 


11-2240054 


•0007072136 


1415 


2002225 


2833148375 


37-6164857 


11-2267007 


-0007067138 


1416 


2005056 


2839159296 


87-6297754 


11-2293448 


-0007062147 


1417 


2007889 


2845178713 


87-6430604 


11-2319876 


-0007057163 


1418 


2010724 


2851206632 


37-6563407 


11-2346292 


•0007052186 


1419 


2013561 


2857243059 


37-6696164 


11-2372696 


•0007047216 


1420 


2016400 


2863288000 


87-6828874 


11-2399087 


-0007042254 


1421 


2019241 


2869341461 ' 


37-6961536 


11-2425465 


•0007037298 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 123 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1422 


2022084 


2875403448 


37-7094153 


11-2451831 


-0007032349 


1423 


2024929 


2881473967 


37-7226722 


11-2478185 


-0007027407 


1424 


2027776 


2887553024 


37-7359245 


11-2504527 


•0007022472 


1425 


2030625 


2893640625 


37-7491722 


11-2530856 


•0007017544 


1426 


2033476 


2899736776 


37-7624152 


11-2557173 


•0007012623 


1427 


2036329 


2905841483 


37-7756535 


11-2583478 


•0007007708 


1428 


2039184 


2911954752 


37-7888873 


11-2609770 


•0007002801 


1429 


2042041 


2918076589 


37-8021163 


11-2636050 


•0006997901 


1430 


2044900 


2924207000 


37-8153408 


11-2662318 


•0006993007 


1431 


2047761 


2930345991 


37-8285606 


11-2688573 


•0006988120 


1432 


2050624 


2936493568 


37-8417759 


11-2714816 


•0006983240 


1433 


2053489 


2942649737 


37-8549864 


11-2741047 


•0006978367 


1434 


2056356 


2948814504 


37-8681924 


11-2767266 


•0006973501 


1435 


2059225 


2954987875 


37-8813938 


11-2793472 


•0006968641 


1436 


2062096 


2961169856 


37-8945906 


11-2819666 


•0006963788 


1437 


2064969 


2967360453 


37-9077828 


11-2845849 


•0006958942 


1438 


2067844 


2973559672 


37-9209704 


11-2872019 


•0006954103 


1439 


2070721 


2979767519 


37-9341538 


11-2898177 


•0006949270 


1440 


2073600 


2985984000 


37-9473319 


11-2924323 


•0006944444 


1441 


2076481 


2992209121 


37-9605058 


11-2950457 


•0006939625 


1442 


2079364 


3098442888 


37-9736751 


11-2976579 


•0006934813 


1443 


2082249 


3004685307 


37-9868398 


11-3002688 


•0006930007 


1444 


2085136 


3010936384 


38-0000000 


11-3028786 


•0006925208 


1445 


2088025 


3017196125 


380131556 


11-3054871 


•0006920415 


1446 


2080916 


3023464536 


38-0263067 


11-3080945 


•0006915629 


1447 


2093809 


3029741623 


38-0394532 


11-3107006 


•0006910850 


1448 


2096704 


3086027392 


38-0525952 


11-3133056 


•0006906078 


1449 


2099601 


3042321849 


38-0657326 


11-3159094 


•0006901312 


1450 


2102500 


3048625000 


38-0788655 


11-3185119 


•0006896552 


1451 


2105401 


3054936851 


88-0919939 


11-3211132 


•0006891799 


1452 


2108304 


3061257408 


88-1051178 


11-8237134 


•0006887052 


1453 


2111209 


3067586777 


38-1182371 


11-3263124 


•0006882312 


1454 


2114116 


3073924664 


38-1313519 


11-3289102 


-0006877579 


1455 


2117025 


3080271375 


88-1444622 


11-3315067 


-0006872852 


1456 


2119936 


3086626816 


38-1575681 


11-3341022 


-0006868132 


1457 


2122849 


3092990993 


38-1706693 


11-3366964 


•0006863412 


1458 


2125764 


3099363912 


38-1837662 


11-3392894 


•0006858711 


1459 


2128681 


3105745579 


88-1968585 


11-3418813 


•0006854010 


1460 


2131600 


3112136000 


38-2099463 


11-3444719 


•0006849315 


1461 


2134521 


3118535181 


38-2230297 


11-8470614 


•0006844627 


1462 


2137444 


3124943128 


88-2361085 


11-3496497 


•0006839945 


1463 


2140369 


3131359847 


38-2491829 


11-3522368 


•0006835270 


1464 


2143296 


3137785344 


38-2622529 


11-3548227 


•0006830601 


1465 


2146225 


3144219625 


38-2753184 


11-3574075 


•0006825939 


1466 


2149156 


3150662696 


38-2883794 


11-3599911 


•0006821282 


1467 


2152089 


3157114563 


38-3014360 


11-3625735 


•0006816633 


1468 


2155024 


3163575232 


88-3144881 


11-3651547 


•0006811989 


1469 


2157961 


3170044709 


38-3275358 


11-3677347 


•0006807352 


1470 


2160900 


3176523000 


38-3405790 


11-3703136 


•0006802721 


1471 


2163841 


3183010111 


38-3536178 


11-3728914 


•0006798097 


1472 


2166784 


3189506048 


88-3666522 


11-3754079 


•0006793478 


1473 


2169729 


3196010817 


38-3796821 


11-3780433 


•0006788866 


1474 


2172676 


3202524424 


38-3927076 


11-8806175 


•0006784261 


1475 


2175625 


3209046875 


88-4057287 


11-8831906 


•0006779661 


1476 


2178576 


3215578170 


38-4187454 


11-3857625 


•0006775068 


1477 


2181529 


3222118333 


38-4317577 


11-8883332 


•0006770481 


1478 


2184484 


3228667352 


38-4447656 


11-3909028 


•0006765900 


1479 


2187441 


3235225239 


88-4577691 


11-3934712 


•0006761325 


1480 


2190400 


3241792000 


38-4707681 


11-3960384 


•0006756757 


1481 


2193361 


3248367641 


88-4837627 


11-3986045 


•0006752194 


1J82 


2196324 


3254952168 


38-4967530 


11-4011695 


•0006747638 


1483 


2199289 


3261545587 


38-5097390 


11-4037332 


•0006743088 



124 



THE PRACTICAL MODEL CALCULATOR. 



Number. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1484 


2202256 


3268147904 


38-5227206 


11-4062959 


•0006738544 


1485 


2205225 


3274759125 


38-5356977 


11-4088574 


•0006734007 


1486 


2208196 


3281379256 


38-5486705 


11-4114177 


-0006729474 


1487 


2211169 


3288008303 


38-5616389 


11-4139769 


-0006724950 


1488 


2214144 


3294646272 


38-5746030 


11-4165349 


-0006720430 


1489 


2217121 


3301293169 


38-5875627 


11-4190918 


-0006715917 


1490 


2220100 


3307949000 


38-6005181 


11-4206476 


-0006711409 


1491 


2223081 


3314613771 


38-6134691 


11-4242022 


-0006706908 


1492 


2226004 


3321287488 


38-6264158 


11-4267556 


-0006702413 


1493 


2229049 


3227970157 


38-6393582 


11-4293079 


-0006697924 


1494 


2232036 


3334661784 


38-6522962 


11-4318591 


-0006693440 


1495 


2235025 


8341362375 


38-6652299 


11-4344092 


•0006688963 


1496 


2238016 


3348071936 


38-6781593 


11-4369581 


-0006684492 


1497 


2241009 


3354790473 


38-6910843 


11-4395059 


-0006680027 


1498 


2244004 


3361517992 


38-7040050 


11-4420525 


-0006675567 


1499 


2247001 


3368254499 


38-7169214 


11-4445980 


-0006671114 


1500 


2250000 


3375000000 


38-7298335 


11-4471424 


-0006666667 


1501 


2253001 


3381754501 


38-7427412 


11-4496857 


•0006662225 


1502 


2256004 


3388518008 


38-7556447 


11-4522278 


-0006657790 


1503 


2259009 


3395290527 


38-7685439 


11-4547688 


-0006553360 


1504 


2262016 


3402072064 


38-7814389 


11-4573087 


-0006648936 


1505 


2265025 


3408862625 


38-7943294 


11-4598476 


-0006644518 


1506 


2268036 


3415662216 


38-8072158 


11-4623850 


-0006640106 


1507 


2271049 


3422470843 


38-8200978 


11-4649215 


-0006635700 


1508 


2274064 


3429288512 


38-8329757 


11-4674568 


-0006631300 


1509 


2277081 


3436115229 


38-8458491 


11-4699911 


-0006626905 


1510 


2280100 


3442951000 


38-8587184 


11-4725242 


-0006622517 


1511 


2283121 


3449795831 


38-8715834 


11-4750562 


•0006618134 


1512 


2286144 


3456649728 


38-8844442 


11-4775871 


•0006613757 


1513 


2289169 


3463512697 


38-8973006 


11-4801169 


-0006609385 


1514 


2292196 


3470384744 


38-9101529 


11-4826455 


-0006605020 


1515 


2295225 


3477265875 


38-9230009 


11-4851731 


-0006600660 


1516 


2298256 


3484156096 


38-9358447 


11-4876995 


-0006596306 


1517 


2301289 


3491055413 


38-9486841 


11-4902249 


-0006591958 


1518 


2304324 


3597963832 


38-9615194 


11-4927491 


•0006587615 


1519 


2307361 


3504881359 


38-9743505 


11-4952722 


-0006583278 


1520 


2310400 


3511808000 


38-9871774 


11-4977942 


-0006578947 


1521 


2313441 


3518743761 


39-0000000 


11-5003151 


•0006574622 


1522 


2316484 


3525688648 


39-0128184 


11-5028348 


•0006570302 


1523 


23] 9529 


3532642667 


39-0256326 


11-5053535 


•0006565988 


1524 


2322576 


3539605824 


39-0384426 


11-5078711 


•0006561680 


1525 


2325625 


3546578125 


39-0512483 


11-5103876 


•0006557377 


1526 


2328676 


3553559576 


39-0640499 


11-5129030 


•0006553080 


1527 


2331729 


3567549552 


39-0768473 


11-5154173 


•0006548788 


1528 


2334784 


3560558183 


39-0896406 


11-5179305 


•0006544503 


1529 


2337841 


3574558889 


39-1024296 


11-5204425 


•0006540222 


1530 


2340900 


3581577000 


39-1152144 


11-5229535 


•0006535948 


1531 


2343961 


3588604291 


39-1279951 


11-5254634 


•0006531679 


1532 


2347024 


3595640768 


39-1407716 


11-5279722 


•0008527415 


1533 


2350089 


3602686437 


39-1535439 


11-5304799 


•0006523157 


1534 


2353156 


3609741304 


39-1663120 


11-5329865 


•0006518905 


1535 


2356225 


3616805375 


39-1790760 


11-5354920 


•0000514658 


1636 


2359256 


3623878656 


39-1918359 


11-5379965 


•0006510417 


1537 


2362369 


3630961153 


39-2045915 


11-5404998 


•0006508181 


1538 


2365444 


3638052872 


39-2173431 


11-5430021 


•0006501951 


1539 


2368521 


3645153819 


39-2300905 


11-5455033 


•0006497726 


1540 


2371600 


3652264000 


39-2428337 


11-5480034 


•0006493506 


1541 


2374681 


3659383421 


39-2555728 


11-5505025 


•0006489293 


1542 


2377764 


3666512088 


39-2683078 


11-5530004 


•0006485084 


1543 


2380849 


3673650007 


39-2810387 


11-5554972 


•0006480881 


1544 


2383936 


3680797184 


39-2937654 


11-5579931 


•0006476684 


1545 


2387025 


3687953625 


39-3084880 


11-5604878 


•0008472492 

1 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



125 



Number. 


Squares. 


Cubes. 


Square Boots. 


Cube Roots. 


Reciprocals. 


1546 


2390116 


3695119336 


39-3192065 


11-5629815 


•0006468305 


1547 


2393209 


3702294323 


39-3319208 


11-5654740 


•0006464124 


1548 


2396304 


3709478592 


39-3446311 


11-5679655 


•0006459948 


154'J 


2399401 


3716672149 


393573373 


11-5704559 


-0006455778 


1550 


2402500 


3723875000 


39-3700394 


11-5729453 


•00U6451613 


1551 


2405601 


3731087151 


39-3827373 


11-5754336 


-0006447453 


1552 


2408704 


3738308608 


39-3954312 


11-5779208 


•0006443299 


1553 


2411809 


3745539377 


39-4081210 


11-5804069 


-0006439150 


1554 


2414916 


3752779464 


39-4208067 


11-5828919 


•0006435006 


1555 


2418025 


3760028875 


39-4334883 


11-5853759 


•0006430868 


1556 


2421136 


3767287616 


39-4461658 


11-5878588 


-0006426735 


1557 


2424249 


3774555693 


39-4588393 


11-5903407 


•0006422608 


1558 


2427364 


3781833112 


39-4715087 


11-5928215 


•0006418485 


1559 


2430481 


3789119879 


39-4841740 


11-5953013 


•0006414368 


1560 


2433600 


3796416000 


39-4968353 


11-5977799 


•0006410256 


1561 


2436721 


3803721481 


39-5094925 


11-6002576 


•0006406150 


1562 


2439844 


3811036328 


39-5221457 


11-6027342 


•0006402049 


1563 


2442969 


3818360547 


39-5347948 


11-6052097 


•0006397953 


1564 


2446096 


3825641444 


39-5474399 


11-6076841 


•0006393862 


1565 


2449225 


3833037125 


39-5600809 


11-6101575 


•0006389776 


1566 


2452356 


3840389496 


39-5727179 


11-6126299 


•0006385696 


1567 


2455489 


3847751263 


39-5853508 


11-6151012 


•0006381621 


1568 


2458624 


3855123432 


39-5979797 


11-6175715 


•0006377551 


1569 


2461761 


3862503009 


39-6106046 


11-6200407 


•0006373488 


1570 


2464900 


3869883000 


39-6232255 


11-6225088 


•0006369427 


1571 


2468041 


3877292411 


39-6358424 


11-6249759 


•0006365372 


1572 


2471184 


3884701248 


39-6484552 


11-6274420 


•0006361323 


1573 


2474329 


3892119157 


39-6610640 


11-6299070 


•0006357279 


1574 


2477476 


3899547224 


39-67366S8 


11-6323710 


•0006353240 


1575 


2480625 


3908984375 


39-6862696 


11-6348339 


•0006349206 


1576 


2483776 


3914430976 


39-6988665 


11-6372957 


•0006345178 


1577 


2486929 


3921887033 


39-7114593 


11-6397566 


•0006341154 


1578 


2490084 


3929352552 


39-7240481 


11-6422164 


•0006337136 


1579 


2493241 


3936827539 


39-7366329 


11-6446751 


•0006333122 


1580 


2496400 


3944312000 


39-7492138 


11-6471329 


•0006329114 


1581 


2499561 


3951805941 


39-7617907 


11-6495895 


•0006325111 


1582 


2502724 


3959309368 


39-7743636 


11-6520452 


•0006321113 


1583 


2505889 


3966822287 


39-7869325 


11-6544998 


•0006317119 


1584 


2509056 


3974344704 


39-7994976 


11-6569534 


•0006313131 


1585 


2512225 


3981876625 


39-8120585 


11-6594059 


•0006309148 


1586 


2515396 


3989418056 


39-8246155 


11-6618574 


•0006305170 


1587 


2518569 


3996969003 


39-8371686 


11-6643079 


-0006301197 


1588 


2521744 


4004529472 


39-8497177 


11-6667574 


•0006297229 


1589 


2524921 


4012099469 


39-8622628 


11-6692058 


•0006293266 


1590 


2528100 


4014679000 


39-8748040 


11-6716532 


•0006289308 


1591 


2531281 


4027268071 


39-8873413 


11-6740996 


•0006285355 


1592 


2534464 


4034866688 


39-8998747 


11-6765449 


•0006281407 


1593 


2537649 


4042474857 


39-9124041 


11-6789892 


•0006277464 


1594 


2540836 


4050092584 


39-9249295 


11-6814325 


-0006273526 


1595 


2544025 


4057719875 


39-9374511 


11-6838748 


-0000269592 


1596 


2547216 


4065356736 


39-9499687 


11-6863161 


•0006265664 


1597 


2550409 


4073003173 


39-9624824 


11-6887563 


-0006261741 


1598 


2553604 


4080659192 


39-9749922 


11-6911955 


-0006257822 


1599 


2556801 


4088324799 


39-9874980 


11-6936337 


•0006253909 


1600 


2560000 


4096000000 


40-0000000 


11-6960709 


•0006250000 



To find the square or cube root of a number consisting of integers 

and decimals. 

Rule. — Multiply the difference between the root of the integer 

part of the given number, and the root of the next higher integer 

number, by the decimal part of the given number, and add the 

L 2 



126 THE PRACTICAL MODEL CALCULATOR. 

product to the root of tlie given integer number ; the sum is the 
root required. 

Ee quired the square root of 20*321. 

Square root of 21 = 4-5825 
Do. 20 = 44721 

•1104 X -321 + 4-4721 = 4-5075384, the 
square root required. 

Required the cube root of 16-42. 
Cube root of 17 = 2-5712 
Do. 16 = 2-5198 

•0514 X -42 + 2-5198 = 2^541388, the cube 
root required. 

To find tlie squares of numhers in arithmetical progression ; or, 
to extend the foregoing tahle of squares. 

Rule. — Find, in the usual way, the squares of the first two num- 
bers, and subtract the less from the greater. Set down the square 
of the larger number, in a separate column, and add to it the dif- 
ference already found, with the addition of 2, as a constant quan- 
tity ; the product will be the square of the next following number. 

The square of 1500 = 2250000 2250000 

The square of 1499 = 2247001 

Di&rence 2999 + 2 = 3001 

The square of 1501 2253001 

Difference 3001 + 2 = 3003 

The square of 1502 2256004 

To find the square of a greater number than is contained in the table. 

Rule 1. — If the number required to be squared exceed by 2, 3, 4, 
or any other number of times, any number contained in the table, 
let the square affixed to the number in the table be multiplied by 
the square of 2, 3, or 4, fcc, and the product will be the answer 
sought. 

Required the square of 2595. 

2595 is three times greater than 865 ; and the square of 865, 
by the table, is 748225. 

Then, 748225 X 3^ = 6734025. 

Rule 2. — If the number required to be squared be an odd num- 
ber, and do not exceed twice the amount of any number contained 
in the table, find the two numbers nearest to each other, which, 
added together, make that sum ; then the sum of the squares of 
these two numbers, by the table, multiplied by 2, will exceed the 
square required by 1. 

Required the square of 1865. 

The two nearest numbers (932 + 933) = 1865. 

Then, by table (932^ = 868624) + (933^ = 870489) = 1739113 x 
2 = 3478226 - 1 = 3478225. 



RULES FOR SQUARES, CUBES, SQUARE ROOTS, ETC. 127 

To find the cube of a greater number than is contained in the table. 

Rule. — Proceed, as in squares, to find liow many times the num- 
ber required to be cubed exceeds the number contained in the table. 
Multiply the cube of that number by the cube of as many times as 
the number sought exceeds the number in the table, and the pro- 
duct will be the answer required. 

Required the cube of 3984. 

3984 is 4 times greater than 996 ; and the cube of 996, by the 
table, is 988047936. 

Then, 988047936 x 4^ = 63235067904. 

To find the square or cube root of a higher number than is in the table. 

Rule. — Refer to the table, and seek in the column of squares 
or cubes the number nearest to that number whose root is sought, 
and the number from which that square ov- cube is derived will be 
the answer required, when decimals are not of importance. 

Required the square root of 542869. 

In the Table of Squares, the nearest number is 543169 ; and 
the number from which that square has been obtained is 737. 
Therefore, ^/542869 = 737 nearly. 

To find more nearly the cube root of a higher numher than is in 

the table. 

Rule. — Ascertain, by the table, the nearest cube number to the 
number given, and call it the assumed cube. 

Multiply the assumed cube, and the given number, respectively, 
by 2 ; to the product of the assumed cube add the given number, 
and to the product of the given number add the assumed cube. 

Then, by proportion, as the sum of the assumed cube is to the 
sum of the given number, so is the root of the assumed cube to 
the root of the given number. 

Required the cube root of 412568555. 

By the table, the nearest number is 411830784, and its cube 
root is 744. 

Therefore, 411830784 x 2 + 412568555 = 1236230123. 
And, 412568555 x 2 + 411830784 = 1236967894. 
Hence, as 1236230123 : 1236967894 : : 744 : 744-369, very nearly. 

To find the square or cube root of a number containing decimals. 

Rule. — Subtract the square root or cube root of the integer of 
the given number from the root of the next higher number, and 
multiply the difference by the decimal part. The product, added to 
the root of the integer of the given number will be the answer 
required. 

Required the square root of 321-62. 

^/321 = 17-9164729, and v/322 = 17-9443584; the difference 
(•0278855) X -62 + 17-9164729 = 17-9337619. 



128 



THE PRACTICAL MODEL CALCULATOR. 



To ohtain the square root or cube root of a number containing deci- 
mals^ by inspection. 
BuLE. — The square or cube root of a number containing deci- 
mals may be found at once by inspection of the tables, by taking 
the figures cut off in the number, by the decimal point, in ^ja^Vs 
if for the square root, and in triads if for the cube root. The fol- 
lowing example will show the results obtained, by simple inspec- 
tion of the tables, from the figures 234, and from the numbers 
formed by the addition of the decimal point or of ciphers. 



Number. 


Square Hoot. 


Cube Root. 


•00234 


•0483735465* 


•132761439t 


•0234 


•152970585 


•2841 


•2340 


•483735465 


•61622401 


2-34 


1-52970585 


1-32761439 


23-40 


4-83735465 


2-860 


234 


15-2970585 


6-1622401 


2340 


' 48-3735465 


13-2761439 


23400 


152-970585 


28-60 



To find the cubes of numbers in arithmetical progression, or to extend 
the preceding table of cubes. 

Rule. — Find the cubes of the first two numbers, and subtract 
the less from the greater. Then, multiply the least of the two 
numbers cubed by 6, add the product, with the addition of 6 as a 
constant quantity, to the difference ; and thus, adding 6 each time 
to the sum last added, form a first series of differences. 

To form a second series of differences, bring down, in a separate 
column, the cube of the highest of the above numbers, and add 
the difference to it. The amount will be the cube of the next 
general number. 

Required the cubes of 1501, 1502, and 1503. 

First series of differences. 
By Tab. 1500 = 3375000000 



1499 = 3368254499 



1499x6+6 = 
9000 + 6 = 
9006 4- 6 = 



6745501 diflference. 
9000 



6754501 cliff, of 1500 
9006 



6763507 diff. of 1501 
9012 



6772519 diff. of 1502 



&c., &c. 



Second series of differences. 

Then, 3375000000 Cube of 1500 
Diff. for 1500 = 6754501 



3381754501 Cube of 1501 
Diff. for 1501 = 6763507 



3388518008 Cube of 1502 
Diff. for 1502= 6772519 



3395290527 Cube of 1503 
&c., &c. 



* Derived from -002340 by means of 2340. 

f Derived from -002340 by means of 2340. 

X The nearest result by simple inspection is obtained for -023 by 23. But four 
places correct can always be obtained by looking in the table of cubes for the 
nearest triad or triads, in this instance for 23400 ; the cube beginning with the 
figures 23393 is that of 2860, whence -2860 is true to the last place, and is after- 
wards substituted. 



TABLE OF THE FOURTH AND FIFTH POWERS OF NUMBERS. 129 



Table of the Fourth and Fifth Powers of Numbers. 



Number. 


4th Power. 


5th Power. 


Number. 


4th Power. 


5th Power. 


1 


1 


1 


76 


33362176 


2535525376 


2 


16 


32 


77 


35153041 


2706784157 


3 


81 


243 


78 


37015056 


2887174368 


4 


256 


1024 


79 


38950081 


3077056399 


5 


625 


3125 


80 


40960000 


3276800000 


6 


1296 


7776 


81 


43046721 


3486784401 


7 


2401 


16807 


82 


45212176 


3707398432 


8 


4096 


32768 


83 


47458321 


3939040643 


9 


6561 


59049 


84 


49787136 


4182119424 


10 


10000 


100000 


85 


52200625 


4437053125 


11 


14641 


161051 


86 


54708016 


4704270176 


12 


20736 


248832 


87 


57289761 


4984209207 


13 


28561 


371293 


88 


59969536 


5277319168 


14 


38416 


537824 


89 


62742241 


5584059449 


15 


50625 


759375 


90 


65610000 


5904900000 


16 


65536 


1048576 


91 


68574961 


6240321451 


17 


83521 


1419857 


92 


716.39296 


6590S15232 


18 


104976 


1889568 


93 


74805201 


6596883693 


19 


130321 


2476099 


94 


78074S96 


7339040224 


20 


160000 


3200000 


95 


81450625 


7737809375 


21 


194481 


4084101 


96 


84934656 


8153726976 


22 


234256 


6153632 


97 


8S529281 


8587340257 


23 


279841 


6436343 


98 


92236S16 


9039207908 


24 


331776 


7962624 


99 


96059601 


9509900499 


25 


390625 


97G5625 


100 


100000000 


10000000000 


26 


456976 


11881376 


101 


104060401 


10510100501 


27 


531441 


14.348907 


102 


108243216 


11040808032 


28 


614656 


17210368 


103 


112550881 


11592740743 


29 


707281 


20511149 


104 


116985856 


12166529024 


30 


810000 


24300000 


105 


121550625 


12762815625 


31 


923521 


28629151 


106 


126247696 


13382255776 


32 


1048576 


33554432 


107 


131079601 


14025517307 


33 


1185921 


39135393 


108 


136048896 


14693280768 


34 


1336336 


45435424 


109 


141158161 


15386239549 


35 


1500625 


62521875 


110 


146410000 


16105100000 


36 


1679616 


60466176 


111 


151807041 


16850581551 


37 


1874161 


69343957 


112 


157351936 


17623416832 


38 


2085136 


79235168 


113 


163047S61 


18424351793 


39 


2313441 


90224199 


114 


168896016 


19254145824 


40 


2560000 


102400000 


115 


174900625 


20113571875 


41 


2825761 


115856201 


116 


181063936 


21003416576 


42 


3111696 


1306912.32 


117 


187388721 


21924480357 


43 


3418801 


147008443 


118 


193877776 


22877577568 


44 


3748096 


164916224 


119 


200533921 


23863536599 


45 


4100625 


184528125 


120 


207360000 


24883200000 


46 


4477456 


205962976 


121 


214358881 


25937424601 


47 


4879681 


229345007 


122 


221533456 


27027081632 


48 


5308416 


2.54803968 


123 


228886641 


28153056843 


49 


5764801 


282475249 


124 


236421376 


29316250624 


50 


6250000 


312500000 


125 


244140625 


30.517578125 


51 


6765201 


345025251 


126 


252047376 


31757969376 


52 


7311616 


380204032 


127 


260144641 


33038369407 


53 


7890481 


418195493 


128 


268435456 


34359738368 


54 


8503056 


459165024 


129 


276922881 


35723051649 


55 


9150625 


503284375 


130 


285610000 


37129300000 


56 


9834496 


550731776 


131 


294499921 


38579489651 


57 


10556001 


601692057 


132 


303595776 


40074642432 


58 


11316496 


656356768 


133 


312900721 


41615795893 


59 


12117361 


714924299 


134 


322417936 


43204003424 


60 


12900000 


777600000 


135 


332150625 


448403.34.375 


61 


13845841 


844596301 


136 


342102016 


46525874176 


62 


14776336 


916132832 


137 


352275361 


48261724457 


63 


15752961 


992436543 


138 


362673936 


60049003168 


64 


16777216 


1073741824 


1.39 


373301041 


61888S44699 


65 


17850625 


1160290625 


140 


384160000 


63782400000 


66 


18974736 


1252.332576 


141 


395254161 


55730836701 


67 


20151121 


1350125107 


142 


406586896 


57735339232 


68 


21381376 


1453933568 


143 


418161601 


59797108943 


69 


22667121 


15640.31349 


144 


429981696 


61917364224 


70 


24010000 


1680700000 


145 


442050625 


64097340625 


71 


25411681 


1804229-351 


146 


454.371856 


66338290976 


72 


26873S56 


1934917632 


147 


466948881 


68641485507 


73 


28398241 


2073071593 


148 


479785216 


7100S211968 


74 


29986576 


2219006624 


149 


492884401 


73439775749 


75 


31640625 


2373046875 


150 


506250000 


75937500000 



130 



THE PRACTICAL MODEL CALCULATOR, 





Table 


of Syperholic Logarithms 






N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


1-01 


•0099503 


1-58 


•4574248 


2-15 


•7654678 


2-72 


1-0006318 


1-02 


•0198026 


1-59 


•4637340 


2-16 


•7701082 


2-73 


1-0043015 


1-03 


•0295588 


1-60 


•4700036 


2^17 


•7747271 


2-74 


1-0079579 


1-04 


•0392207 


1-61 


•4762341 


2-18 


•7793248 


2-75 


1-0116008 


1-05 


•0487902 


1-62 


•4824261 


2-19 


•7839015 


2-76 


1-0152306 


1-06 


•0582689 


1-63 


•4885800 


2-20 


•7884573 


2-77 


1-0188473 


1-07 


•0676586 


1^64 


•4946962 


2-21 


•7929925 


2-78 


1-0224509 


1-08 


•0769610 


1^65 


•5007752 


2''^2 


•7975071 


2-79 


1^026041 


1-09 


•0861777 


1-66 


•5068175 


2^23 


•8020015 


2-80 


1-0296194 


1-10 


•0953102 


r67 


•5128236 


2-24 


•8064758 


2-81 


1-0331844 


1-11 


•1043600 


1^68 


•5187937 


2-25 


•8109302 


2-82 


1-0367368 


1-12 


•1133287 


1-69 


•5247285 


2-26 


•8153648 


2-83 


1-0402766 


1-13 


•1222176 


1^70 


•5306282 


2-27 


•8197798 


2-84 


1-0438040 


1-14 


•1310283 


1-71 


•5364933 


2-28 


•8241754 


2-85 


1 •0473189 


1-15 


•1397619 


1-72 


•5423242 


2-29 


•8285518 


2-86 


1-0508216 


1-16 


•1484200 


1-73 


•5481214 


2-30 


•8329091 


2-87 


1-0543120 


1-17 


•1570037 


1-74 


•5538851 


2-31 


•8372475 


2-88 


1-0577902 


MB 


•1655144 


1-75 


•5596157 


2-32 


•8415671 


2-89 


1-0612564 


1-19 


•1739533 


1-76 


•5653138 


2-33 


•8458682 


2-90 


1-0647107 


1-20 


•1823215 


1-77 


•5709795 


2-34 


•8501509 


2-91 


1-0681530 


1-21 


•1906203 


1^78 


•6766133 


2-35 


•8544153 


2-92 


1-0715836 


1-22 


•1988508 


1^79 


•5822156 


2-36 


•8586616 


2-93 


1-0750024 


1-23 


•2070141 


1-80 


•5877866 


2-37 


•8628899 


2-94 


1-0784095 


1-24 


•2151113 


1^81 


•5933268 


2-38 


•8671004 


2-95 


1-0818051 


1-25 


•2231435 


1-82 


•5988365 


2-39 


•8712933 


2-96 


1-0851892 


1-26 


•2311117 


1-83 


•6043159 


2-40 


•8754687 


2-97 


1-0885619 


1-27 


•2390169 


1-84 


•6097655 


2^41 


•8796267 


2-98 


1-0919233 


1-28 


•2468600 


1-85 


•6151856 


2^42 


•8837675 


2-99 


1-0952733 


1-29 


•2546422 


1-86 


•6205764 


2^43 


•8878912 


3-00 


1-0986123 


1-30 


•2623642 


1-87 


•6259384 


£•44 


•8919980 


3-01 


1-1019400 


1-31 


•2700271 


1-88 


•6312717 


2^45 


•8960880 


3-02 


1-1052568 


1-32 


•2776317 


1-89 


•6365768 


2^46 


•9001613 


3-03 


1-1085626 


1-33 


•2851789 


1-90 


■6418538 


2-47 


•9042181 


3-04 


1-1118575 


1-34 


•2926696 


1-91 


•6471032 


2^48 


•9082585 


3-05 


1-1151415 


1-35 


•3001045 


1-92 


•6523251 


2^49 


•9122826 


3-06 


1-1184149 


1-36 


•3074846 


1-93 


♦6575200 


2-50 


•9162907 


3-07 


1-1216775 


1-37 


•3148107 


1-94 


•6626879 


2-51 


•9202827 


3-08 


1-1249295 


1-38 


•3220834 


1-95 


•6678293 


2^52 


•9242589 


3-09 


1-1281710 


1-39 


•3293037 


1-96 


•6729444 


2-53 


•9282193 


3-10 


1-1314021 


1-40 


•3364722 


1-97 


•6780335 


2-54 


•9321640 


3-11 


1-1346227 


1-41 


•3435897 


1-98 


•6830968 


2^55 


•9360933 


S-12 


1-1378330 


1-42 


•3506568 


1-99 


•6881346 


2-56 


•9400072 


3-13 


1-1410330 


1-43 


•3576744 


2-00 


•6931472 


2-57 


•9439058 


3-14 


1-1442227 


1-44 


•3646431 


2-01 


•6981347 


2-58 


•9477893 


3-15 


1-1474024 


1-45 


•3715635 


2-02 


•7030974 


2-59 


■9516578 


3-16 


1-1505720 


1-46 


•3784364 


2-03 


•7080357 


2-60 


•9555114 


3-17 


M537315 


1-47 


•3852624 


2 04 


•7129497 


2-61 


•9593502 


3^18 


1-1568811 


1-48 


•3920420 


2-05 


•7178397 


2-62 


•9631743 


3-19 


1^1600209 


1-49 


•3987761 


2-06 


•7227059 


2-63 


•9669838 


3-20 


1-1631508 


1-50 


•4054651 


2-07 


•7275485 


2-64 


•9707789 


3-21 


1-1662709 


1-51 


•4121096 


2-08 


•7323678 


2-65 


•9745596 


3-22 


1-1693813 


1-52 


•4187103 


2-09 


•7371640 


2-66 


•9783261 


3-23 


1-1724821 


1-53 


•4252677 


2-10 


•7419373 


2-67 


•9820784 


3^24 


1-1755733 


1-54 


•4317824 


2-11 


•7466879 


2-68 


•9858167 


3-25 


1-1786549 


1-55 


•4382549 


2-12 


•7514160 


2-69 


•9895411 


3-26 


1-1817271 


1-56 


•4446858 


1 2-13 


•7561219 


2-70 


•9932517 


3-27 


1-1847899 


1-57 


•4510756 


1 2-14 


•7608058 


2-71 


•9969486 


3-28 

1 


1-1878434 



TABLE or HYPERBOLIC LOGARITHMS. 



131 



N. 


Logarithm. 


N. 


Logarithm, 


N. 


Logarithm. 


N. 


Logarithm. 


3-29 


1-1908875 


3-91 


1-3635373 


4-53 


1-5107219 


5-15 


1-6389067 


3-30 


1-1939224 


3-92 


1-3660916 


4-54 




5129269 


5-16 


1-6409365 


3-31 


1-1969481 


3-93 


1-3686394 


4-55 




5151272 


5-17 


1-6428726 


3-32 


1-1999647 


3-94 


1-3711807 


4-56 




5173226 


5-18 


1-6448050 


3-33 


1-2029722 


3-95 


1-3737156 


4-57 




5195132 


6-19 


1-6467336 


8-34 


1-2059707 


3-96 


1-3762440 


4-68 




5216990 


5-20 


1-6486586 


3-35 


1-2089603 


3-97 


1-3787661 


4-59 




5238800 


5-21 


1-6505798 


3-36 


1-2119409 


3-98 


1-3812818 


4-60 




5260563 


5-22 


1-6524974 


3-37 


1-2149127 


3-99 


1-3837912 


4-61 




5282278 


5-23 


1-6544112 


3-38 


1-2178757 


4-00 


1-3862943 


4-62 




5303947 


5-24 


1-6563214 


3-39 


1-2208299 


4-01 


1-3887912 


4-63 




5325568 


5-25 


1-6582280 


3-40 


1-2237754 


402 


1-3912838 


4-64 




5347143 


5-26 


1-6601310 


3-41 


1-2267122 


4-03 


1-3937663 


4-65 




5368672 


5-27 


1-6620303 


3-42 


1-2296405 


4 04 


1-3962446 


4-66 




5390154 


6-28 


1-6639260 


3-43 


1-2325605 


4-05 


1-3987168 


4-67 




-5411590 


6-29 


1-6658182 


3-44 


1-2354714 


4-06 


1-4011829 


4-68 




5432981 


5-30 


1-6677068 


3-45 


1-2383742 


4-07 


1-4036429 


4-69 




-5454325 


5-31 


1-6695918 


8-46 


1-2412685 


4-08 


1-4060969 


4-70 




5475625 


5-32 


1-6714733 


3-47 


1-2441545 


4-09 


1-4085449 


4-71 




5496879 


6-33 


1-6733512 


3-48 


1-2470322 


4-10 


1-4109869 


4-72 




•5518087 


5-34 


1-6752256 


3-49 


1-2499017 


4-11 


1-4134230 


4-73 




5539252 


5-35 


1-6770965 


3-50 


1-2527629 


4-12 


1-4158531 


4-74 




5560371 


5-36 


1-6789639 


3-51 


1-2556160 


4-13 


1-4182774 


4-75 




5581446 


5-37 


1-6808278 


3-52 


1-2584609 


4-14 


1-4206957 


4-76 




5602476 


6-38 


1-6826882 


3-63 


1-2612978 


4-15 


1-4231083 


4-77 




5623462 


5-39 


1-6845453 


3-54 


1-2641266 


4-16 


1-4255150 


4-78 




5644405 


5-40 


1-6863980 


3-55 


1-2669475 


4-17 


1-4279160 


4-79 




5665304 


5-41 


1-6882491 


3-56 


1-2697605 


4-18 


1-4303112 


4-80 




5686159 


5-42 


1-6900958 


3-57 


1-2725655 


4-19 


1-4327007 


4-81 




5706971 


5-43 


1-6919391 


3-58 


1-2753627 


4-20 


1-4350845 


4-82 




5727739 


5-44 


1-6937790 


3-59 


1-2781521 


4-21 


1-4374626 


4-83 




5748464 


6-45 


1-6956155 


3-60 


1-2809338 


4-22 


1-4398351 


4-84 




5769147 


6-46 


1-6974487 


3-61 


1-2837077 


4-23 


1-4422020 


4-85 




5789787 


5-47 


1-6992786 


3-62 


1-2864740 


4-24 


1-4445632 


4-86 




5810384 


5-48 


1-7011051 


3-63 


1-2892326 


4-25 


1-4469189 


4-87 




5830939 


6-49 


1-7029282 


3-64 


1-2919836 


4-26 


1-4492691 


4-88 




5851452 


5-50 


1-7047481 


3-65 


1-2947271 


4-27 


1-4516138 


4-89 




5871923 


5-51 


1-7065646 


3-66 


1-2974631 


4-28 


1-4539530 


4-90 




5892352 


6-52 


1-7083778 


3-67 


1-3001916 


4-29 


1-4562867 


4-91 




5912739 


5-53 


1-7101878 


3-68 


1-3029127 


4-30 


1-4586149 


4-92 




5933085 


5-54 


1.7119944 


3-69 


1-3056264 


4-31 


1-4609379 


4-93 




5953389 


5-55 


1-7137979 


3-70 


1-3083328 


4-32 


1-4632553 


4-94 




5973053 


5-56 


1-7155981 


3-71 


1-3110318 


4-33 


1-4655675 


4-95 




5993875 


5-57 


1-7173950 


3-72 


1-3137236 


4-34 


1-4678743 


4-96 




6014057 


5-58 


1-7191887 


3-73 


1-3164.082 


4-35 


1-4701758 


4-97 




6034198 


6-59 


1-7209792 


3-74 


1-3190856 


4-36 


1-4724720 


4-98 




6054298 


6-60 


1-7227666 


3-75 


1-3217558 


4-37 


1-4747630 


4-99 




6074358 


5-61 


1-7245507 


3-76 


1-3244189 


4-38 


1-4770487 


5-00 




6094379 


5-62 


1-7263316 


3-77 


1-3270749 


4-39 


1-4793292 


5-01 




6114359 


5-63 


1-7281094 


3-78 


1-3297240 


4-40 


1-4816045 


5-02 




6134300 


6-64 


1-7298840 


3-79 


1-3323660 


4-41 


1-4838746 


5-03 


I 


6154200 


5-65 


1-7316555 


3-80 


1-3350010 


4-42 


1-4861396 


5-04 




6174000 


5-66 


1-7334238 


3-81 


1-3376291 


4-43 


1-4883995 


5-05 


I 


6193882 


6-67 


1-7351891 


3-82 


1-3402504 


4-44 


1-4906543 


5-06 


I 


6213664 


5-68 


1-7369512 


3-83 


1-3428648 


4-45 


1-4929040 


5-07 


I 


6233408 


5-09 


1-7387102 


3-84 


1-3454723 


4-46 


1-4951487 


5-08 




6253112 


5-70 


1-7404661 


3-85 


1-3480731 


4-47 


1-4973883 


5-09 




6272778 


6-71 


1-7422189 


3-8G 


1-3506671 


4-48 


1-4996230 


5-10 




6292405 


6-72 


1-7439687 


3-87 


1-3532544 


4-49 


1-5018527 


5-11 




6311994 


5-73 


1-7457155 


3-88 


1-3558351 


4-50 


1-5040774 


5-12 




6331544 


5-74 


1-7474591 


3-89 


1-3584091 


4-51 


1-5062971 


5-13 




6351056 


5-75 


1-7491998 


3-90 


1-3609765 


4-52 


1-5085119 


5-14 


1-6370530 1 


5-76 


1-7509374 



132 



THE PRACTICAL MODEL CALCULATOR. 



N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


N. 


Logarithm. 


5-77 


1-7526720 


6-39 


1-8547342 


7-01 


1-9473376 


7-63 


2-0320878 


5-78 


1 •7544036 


6-40 


1-8562979 


7-02 


1-9487632 


7-64 


2-0333976 


5-79 


1-7561323 


6-41 


1-8578592 


7-03 


1-9501866 


7-65 


2-0347056 


5-80 


1-7578579 


6-42 


1-8594181 


7-04 


1-9516080 


7-66 


2-0360119 


5-81 


1-7595805 


6-43 


1-8609745 


7-05 


1-9530275 


7-67 


2-0373166 


5-82 


1-7613002 


6-44 


1-8625285 


7-06 


1-9544449 


7-68 


2-0386195 


5-83 


1-7630170 


6-45 


1-8640801 


7-07 


1-9558604 


7-69 


2-0399207 


5-84 


1-7647308 


6-46 


1-8656293 


7-08 


1-9572739 


7-70 


2-0412203 


5-85 


1-7064416 


6-47 


1-8671761 


7-09 


1-9586853 


7-71 


2-0425181 


5-86 


1-7681496 


6-48 


1-8687205 


7-10 


1-9600947 


7-72 


2-0438143 


5-87 


1-7698546 


6-49 


1-8702625 


7-11 


1-9615022 


7-73 


2-0451088 


5-88 


1-7715567 


6-50 


1-8718021 


7-12 


1-9629077 


7-74 


2-0464016 


5-89 


1-7732559 


6-51 


1-8733394 


7-13 


1-9643112 


7-75 


2-0476928 


6-90 


1-7749523 


6-52 


1-8748743 


7-14 


1-9657127 


7-76 


2-0489823 


5-91 


1-7766458 


6-53 


1-8764069 


7-15 


1-9671123 


7-77 


2-0502701 


5-92 


1-7783364 


6-54 


1-8779371 


7-16 


1-9685099 


7-78 


2-0516563 


5-93 


1-7800242 


6-55 


1-8794650 


7-17 


1-9699056 


7-79 


2-0528408 


5-94 


1-7817091 


6-56 


1-8809906 


7-18 


1-9712993 


7-80 


2-0541237 


5-95 


1-7833912 


6-57 


1-8825138 


7-19 


1-9726911 


7-81 


2-0554049 


5-96 


1-7850704 


6-58 


1-8840347 


7-20 


1-9740810 


7-82 


2-0566845 


5-97 


1-7867469 


6-59 


1-8855533 


7-21 


1-9754689 


7-83 


2-0579624 


5-98 


1-7884205 


6-60 


1-8870696 


7-22 


1-9768549 


7-84 


2-0592388 


5-99 


1-7900914 


6-61 


1-8885837 


7-23 


1-9782390 


7-85 


2-0605135 


6-00 


1-7917594 


6-62 


1-8900954 


7-24 


1-9796212 


7-86 


2-0617866 


6-01 


1-7934247 


6-63 


1-8916048 


7-25 


1-9810014 


7-87 


2-0630580 


6-02 


1-7950872 


6-64 


1-8931119 


7-26 


1-9823798 


7-88 


2-0643278 


6-03 


1-7967470 


6-65 


1-8946168 


7-27 


1-9837562 


7-89 


2-0655961 


6-04 


1-7984040 


6-66 


1-8961194 


7-28 


1-9851308 


7-90 


2-0668627 


6-05 


1-8000582 


6-67 


1-8976198 


7-29 


1-9865035 


7-91 


2-0681277 


6-06 


1-8017098 


6-68 


1-8991179 


7-30 


1-9878743 


7-92 


2-0693911 


6-07 


1-8033586 


6-69 


1-9006138 


7-31 


1-9892432 


7-93 


2-0706530 


6-08 


1-8050047 


6-70 


1-9021075 


7-32 


1-9906103 


7-94 


2-0719132 


6-09 


1-8066481 


6-71 


1-9035989 


7-33 


1-9919754 


7-95 


2-0731719 


6-10 


1-8082887 


6-72 


1-9050881 


7-34 


1-9933387 


7-96 


2-0744290 


6-11 


1-8099267 


6-73 


1-9065751 


7-35 


1-9947002 


7-97 


2-0756845 


6-12 


1-8115621 


6-74 


1-9080600 


7-36 


1-9960599 


7-98 


2-0769384 


6-13 


1-8131947 


6-75 


1-9095425 


7-37 


1-9974177 


7-99 


2-0781907 


6-14 


1-8148247 


6-76 


1-9110228 


7-38 


1-9987730 


8-00 


2-0794415 


6-15 


1-8164520 


6-77 


1-9125011 


7-39 


2-0001278 


8-01 


2-0806907 


6-16 


1-8180767 


6-78 


1-9139771 


7-40 


2-0014800 


8-02 


2-0819384 


6-17 


1-8196988 


6-79 


1-9154509 


7-41 


2-0028305 


8-03 


2-0831845 


6-18 


1-8213182 


6-80 


1-9169226 


7-42 


2-0041790 


8-04 


2-0844290 


6-19 


1-8229351 


6-81 


1-9183921 


7-43 


2-0055258 


8-05 


2-0856720 


6-20 


1-8245493 


6-82 


1-9198594 


7-44 


2-0068708 


8-06 


2-0869135 


6-21 


1-8261608 


6-83 


1-9213247 


7-45 


2-0082140 


8-07 


2-0881534 


6'22 


1-8277699 


6-84 


1-9227877 


7-46 


2-0095553 


8-08 


2-0893918 


6-23 


1-8293763 


6-85 


1-9242486 


7-47 


2-0108949 


8-09 


2-0906287 


6-24 


1-8309801 


6-86 


1-9257074 


7-48 


2-0122327 


8-10 


2-0918640 


6-25 


1-8325814 


6-87 


1-9271641 


7-49 


2-0135687 


8-11 


2-0930984 


6-26 


1-8341801 


6-88 


1-9286186 


7-50 


2-0149030 


8-12 


2-0943306 


6-27 


1-8357763 


6-89 


1-9300710 


7-51 


2-0162354 


8-13 


2-0955613 


6-28 


1-8373699 


6-90 


1-9315214 


7-52 


2-0175661 


8-14 


2-0967905 


6-29 


1-8389610 


6-91 


1-9329696 


7-53 


2-0188950 


8-15 


2-0980182 


6-30 


1-8405496 


6-92 


1-9344157 


7-54 


2-0202221 


8-16 


2-0992444 


6-31 


1-8421356 


6-93 


1-9358598 


7-55 


2-0215475 


8-17 


2-1004691 


6-32 


1-8437191 


6-94 


1-9373017 


7-56 


2-0228711 


8-18 


2-1016923 


6-33 


1-8453002 


6-95 


1-9387416 


7-57 


2-0241929 


8-19 


2-1029140 


6-34 


1-8468787 


6-96 


1-9401794 


7-58 


2-0255131 


8-20 


2-1041341 


6-35 


1-8484547 


6-97 


1-9416152 


7-59 


2-0268315 


8-21 


2-1053529 


6-36 


1-8500283 


6-98 


1-9430489 


7-60 


2-0281482 


8-22 


2-1065702 


6-37 


1-8515994 


0-99 


1-9444805 


7-61 


2-0294631 


8-23 


2-1077861 


6-38 


1-8531680 


7-00 


1-9459101 


7-62 


2-0307763 


8-24 


2-1089998 



TABLE OF HYPERBOLIC LOGARITHMS. 



133 



1 

N. 


Logarithm. 


N. 


Logarithm. 


.. 


Logarithm. 


N. 


Logarithm. 


8-25 


2-1102128 


8-69 


2-1621729 


9-13 


2-2115656 


9-57 


2-2586332 


8-26 


2-1114243 


8-70 


2-1633230 


9-14 


2-2126603 


9-58 


2-2596776 


8-27 


2-1126343 


8-71 


2-1644718 


9-15 


2-2137538 


9-59 


2-2607209 


8-28 


2-1138428 


8-72 


2-1656192 


9-16 


2-2148461 


9-60 


2-2617631 


8-29 


2-1150499 


8-73 


2-1667653 


9-17 


2-2159372 


9-61 


2-2628042 


8-30 


2-1162555 


8-74 


2-1679101 


9-18 


2-2170272 


9-62 


2-2638442 


8-31 


2-1174596 


8-75 


2-1690536 


9-19 


2-2181160 


9-63 


2-2648832 


8-32 


2-1186622 


8-76 


2-1701959 


9-20 


2-2192034 


9-64 


2-2659211 


8-33 


2-1198634 


8-77 


2-1713367 


9-21 


2-2202898 


9-65 


2-2669579 


8-34 


2-1210632 


8-78 


2-1724763 


9-22 


2-2213750 


9-66 


2-2679936 


8-35 


2-1222615 


8-79 


2-1736146 


9-23 


2-2224590 


9-67 


2-2690282 


8-36 


2-1234584 


8-80 


2-1747517 


9-24 


2-2235418 


9-68 


2-2700618 


8-37 


2-1246539 


8-81 


2-1758874 


9-25 


2-2246235 


9-69 


2-2710944 


8-38 


2-1258479 


8-82 


2-1770218 


9-26 


2-2257040 


9-70 


2-2721258 


8-39 


2-1270405 


8-83 


2-1781550 


9-27 


2-2267833 


9-71 


2-2731562 


8-40 


2-1282317 


8-84 


2-1792868 


9-28 


2-2278615 


9-72 


2-2741856 


8-41 


2-1294214 


8-85 


2-1804174 


9-29 


2-2289385 


9-73 


2-2752138 


8-42 


2-1306098 


8-86 


2-1815467 


9-30 


2-2300144 


9-74 


2-2762411 


8-43 


2-1317967 


8-87 


2-1826747 


9-31 


2-2310890 


9-75 


2-2772673 


8-44 


2-1329822 


8-88 


2-1838015 


9-32 


2-2321626 


9-76 


2-2782924 


8-45 


2-1341664 


8-89 


2-1849270 


9-33 


2-2332350 


9-77 


2-2793165 


8-46 


2-1353491 


8-90 


2-1860512 


9-34 


2-2343062 


9-78 


2-2803395 


8-47 


2-1365304 


8-91 


2-1871742 


9-35 


2-2353763 


9-79 


2-2813614 


8-48 


2-1377104 


8-92 


2-1882959 


9-36 


2-2364452 


9-80 


2-2823823 


8-49 


2-1388889 


8-93 


2-1894163 


9-37 


2-2375130 


9-81 


2-2834022 


8-50 


2-1400661 


8-94 


2-1905355 


9-38 


2-2385797 


9-82 


2-2844211 


8-51 


2-1412419 


8-95 


2-1916535 


9-39 


2-2396452 


9-83 


2-2854389 


8-52 


2-1424163 


8-96 


2-1927702 


9-40 


2-2407096 


9-84 


2-2864556 


8-53 


2-1435893 


8-97 


2-1938856 


9-41 


2-2417729 


9-85 


2-2874714 


8-54 


2-1447609 


8-98 


2-1949998 


9-42 


2-2428350 


9-86 


2-2884861 


8-55 


2-1459312 


8-99 


2-1961128 


9-43 


2-2438960 


9-87 


2-2894998 


8-56 


2-1471001 


9-00 


2-1972245 


9-44 


2-2449559 


9-88 


2-2905124 


8-57 


2-1482676 


9-01 


2-1983350 


9-45 


2-2460147 


9-89 


2-2915241 


8-58 


2-1494339 


9-02 


2-1994443 


9-46 


2-2470723 


9-90 


2-2925347 


8-59 


2-1505987 


9-03 


2-2005523 


9-47 


2-2481288 


9-91 


2-2635443 


8-60 


2-1517622 


9-04 


2-2016591 


9-48 


2-2491843 


9-92 


2-2945529 


8-61 


2-1529243 


9-05 


2-2027647 


9-49 


2-2502386 


9-93 


2-2955604 


8-62 


2-1540851 


9-06 


2-2038691 


9-50 


2-2512917 


9-94 


2-2965670 


8-63 


2-1552445 


9-07 


2-2049722 


9-51 


2-2523438 


9-95 


2-2975725 


8-64 


2-1564026 


9-08 


2-2060741 


9-52 


2-2533948 


9-96 


2-2985770 


8-65 


2-1575593 


9-09 


2-2071748 


9-53 


2-2544446 


9-97 


2-2995806 


8-66 


2-1587147 


9-10 


2-2082744 


9-54 


2-2554934 


9-98 


2-3005831 


8-67 


2-1598687 


9-11 


2-2093727 


9-55 


2-2565411 


9-99 


2-3015846 


8-68 


2-1610215 


9-12 


2-2104697 


9-56 


2-2575877 


10-00 


2-3025851 



Logarithms were invented by Juste Byrge, a Frenchman, and 
not by Napier. See "Biographie Universelle," " The Calculus of 
Form," article 822, and "The Practical, Short, and Direct Method 
of Calculating the Logarithm of any given Number and the Number 
corresponding to any given Logarithm," discovered by 01ive»r Byrne, 
the author of the present work. Juste Byrge also invented the 
proportional compasses, and was a profound astronomer and ma- 
thematician. The common Logarithm of a number multiplied by 
2'302585052994 gives the hyperbolic Logarithm of that number. 
The common Logarithm of 2-22 is -346353 .-. 2-302585 X -346353 
= -7975071 the hyperbolic Logarithm. The application of Loga- 
rithms to the calculations of the Engineer will be treated of here- 
after. 

M 



134 THE PRACTICAL MODEL CALCULATOR. 

COMBIXATIOXS OF ALGEBRAIC QUAXTITIES. 

The following practical examples will serve to illustrate the 
method of combining or representing numbers or quantities alge- 
braically ; the chief object of which is, to help the memory with 
respect to the use of the signs and letters, or si/mhoh. 

Let a = 6, h = 4, c = S, d = 2, e -= 1, and/ = 0. 
Then will, (l)2a + b = 12+4 = 16. 

(2) ah + 2c -d = 24: + 6-2 = 28. 

(3) a^-P + e +f= 36 - 16 + 1-1- = 21. 

(4) 5^ X (a - 6) = 16 X (6 - 4) = 16 X 2 = 32. 

(5) Sahe - 7de = 216 - 14 = 202. 

(6) 2{a - h) (5e - 2d) = (12 - 8) x (15 - 4) = 44. 

c^ — e^ 9 — 1 

(T) ^^jTJ x{a-c) = ^^^ X (6 - 3) = 4 X 3 = 12. 

(8) ^/ (a^ - 2b') -f c^ -/ = v/ (36 - 32) -h 2 - = 4. 
{9) Sab- {a-b- c + d) = 72-1 = 71. 
(10) Sab - {a - b - c - d) = n i- S = 75. 

(^1) ^ l^ab-M) X (^ + ^)=VW^)'' ^^ ^ -^ = '^- 

In solving the following questions, the letters a, b, c, &c. are 
supposed to have the same values as before, namely, 6, 4, 3, &c. ; 
but any other values might have been assigned to them ; therefore, 
do not suppose that a must necessarily be 6, nor that b must be 4, 
for the letter a may be put for any known quantity, number, or 
magnitude whatever ; thus a may represent 10 miles, or 50 pounds, 
or any number or quantity, or it may represent 1 globe, or 2 cuhie 
feet, &c. ; the same may be said of b, or any other letter. 



(1) a + 6 - e = T. (6) 4 {a- - b^) (e - e) = 160. 

(2) nbo- d + e = 35. (7) "' ~ ^' X (c? + c^) = 52. 

(4) X X (J - c + (?) = 2T. (9) 4a^A - (c=^ - (Z - e) = 570. 

(5) 5<.'^d - «^ + Me = 62. (10) ^^lod'-icd) ^ d ~ ''' = "^^ 



In the use of algebraic symbols, 3 -^ 4a — 5 signifies the same 
1. 
thing as 3 (4a — b)^. 



4 ((? -h d)^ {a -f 5)^, or 4 X c -f c^^ X a -f 6^, signifies the 
same thing 2^s 4: ^/ c -{- d ' ^ a -\- b. 



135 



THE STEAM ENGINE. 



The particular example which we shall select is that of an 
engine having 8 feet stroke and 64 inch cylinder. 

The breadth of the web of the crank at the paddle centre is the 
breadth which the web would have if it were continued to the paddle 
centre. Suppose that we wished to know the breadth of the web of 
crank of an engine whose stroke is 8 feet and diameter of cylinder 64 
inches. The proper breadth of the web of crank at paddle centre 
would in this case be about 18 inches. 

To find the hreadtJi of crank at paddle centre. — Multiply the 
square of the length of the crank in inches by 1'561, and then 
multiply the square of the diameter of cylinder in inches by -1235 ; 
multiply the square root of the sum of these products by the square 
of the diameter of the cylinder in inches ; divide the product by 
45 ; finally extract the cube root of the quotient. The result is 
the breadth of the web of crank at paddle centre. 

Thus, to apply this rule to the particular example which we have 
selected, we have 

48 = length of crank in inches. 

48 



2304 
1*561 = constant multiplier. 

3596-5 
505-8 found below. 



4102-3 



64 = diameter of cylinder. 
64 



4096 
•1235 = constant multiplier. 

505-8 



and n/4102-3 = 64-05 nearly. 

4096 = square of the diameter of the cylinder, 

45 ) 262348-5 

5829-97 



and ^5829-97 = 18 nearly. 

Suppose that we wished the proper thickness of the large eye of " 
crank for an engine whose stroke is 8 feet and tliameter of cylinder 
64 inches. The proper thickness for the large eye of crank id 
5-77 inches. 



136 THE PRACTICAL MODEL CALCULATOR. 

EuLE. — To find the thickness of large eye of crank. — Multiply tEe 
square of the length of the crank in inches by 1-561, and then mul- 
tiply the square of the diameter of the cylinder in inches by -1235 ; 
multiply the sum of these products by the square of the diameter of 
the cylinder in inches ; afterwards, divide the product by 1828*28 ; 
divide this quotient by the length of the crank in inches ; finally 
extract the cube root of the quotient. The result is the proper 
thickness of the large eye of crank in inches. 

Thus, to apply this rule to the particular example which we have 
selected, we have 

48 = length of crank in inches. 

48 

2304 
1-561 constant multiplier. 

3596-5 
505-8 



4102-3 



64 = diameter of cylinder in inches. 
64 ^ 



4096 

•1235 = constant multiplier. 

505-8 

4102-3 
4096 = square of diameter. 

48 ) 16803020-8 

1828-28 ) 350062-94 

191-4T 



and ^191-47 = 5-77 nearly. 

The proper thickness of the web of crank at paddle shaft centre 
is the thickness which the web ought to have if continued to centre 
of the shaft. Suppose that it were required to find the proper 
thickness of web of crank at shaft centre for an engine whose 
stroke is 8 feet and diameter of cylinder 64 inches. The proper 
thickness of the web at shaft centre in this case would be 8-97 
inches. 

Rule. — To find tlie thickness of the weh of crank at j^addle shaft 
centre. — Multiply the square of the length of crank in inches by 
1*561, and then multiply the square of the diameter in inches by 
'1235 ; multiply the square root of the sum of these products by the 
.square of the diameter of the cylinder in inches ; divide this quotient 
by 360 ; finally extract the cube root of the quotient. The result is 
the thickness of theVeb of crank at paddle shaft centre in inches. 

Thus, to apply the rule to the particular example Vy'hich we have 
selected, we have 



THE STEAM ENGINE. 187 



48 : 

48 


= length of crank in inches. 




2304 
1-561 : 


= constant multiplier. 




3596-5 
505-8 






4102-3 






64 = 
64 


diameter of cylinder. 




4096 
•1235 = 


constant multiplier. 




505-8 
And v/ 4102-3 = 64-05 nearly. 

4096 = square 


of diameter. 




360) 262348-5 






728-75 
And ^ 782-75 = 9 nearly. 



Suppose that it were required to find the proper diameter for 
the paddle shaft journal of an engine whose stroke is 8 feet and 
diameter of cylinder 64 inches. The proper diameter of the 
paddle shaft journal in this case is 14-06 inches. 

Rule. — To find the diameter of the paddle shaft journal. — Mul- 
tiply the square of the diameter of cylinder in inches by the length 
of the crank in inches ; extract the cube root of the product ; 
finally multiply the result by -242. The final product is the diame- 
ter of the paddle shaft journal in inches. 

Thus, to apply this rule to the particular example which we have 
before selected, we' have 

64 = diameter of cylinder in inches. 
64 

4096 

48 = length of crank in inches. 



196608 



and -^196608 = 58-148 
but 58-148 X -242 = 14-07 inches. 

Suppose it were required to find the proper length of the paddle 
shaft journal for an engine whose stroke is 8 feet, and diameter of 
cylinder 64 inches. The proper length of the paddle shaft journal 
would be, in this case, 17-59 inches. 

The following rule serves for engines of all sizes : 

Rule. — To find the length of the paddle shaft journal. — Multiply 

the square of the diameter of the cylinder in inches by the length 

of the crank in inches ; extract the cube root of the quotient ; 

multiply the result by -303. The product is the length of the 



138 THE PRACTICAL MODEL CALCULATOR. 

paddle shaft journal in inches. (The length of the paddle shaft 
journal is IJ times the diameter.) 

To apply this rule to the example which we have selected, we have 

64 = diameter of cylinder in inches. 
64 



4096 

48 = length of crank in inches. 



196608 



and ^ 196608 = 58-148 
.-. length of journal = 58-148 X -303 = 17-60 inches. 

"\Ye shall now calculate the proper dimensions of some of those 
parts which do not depend upon the length of the stroke. Suppose 
it were required to find the proper dimensions of the respective parts 
of a marine engine the diameter of whose cylinder is 64 inches. 

Diameter of crank-pin journal = 90-9 inches, or ahout 9 inches. 

Length of crank-pin journal = 10-18 inches, or nearly 10| 
inches. 

Breadth of the eye of cross-head = 2*64 inches, or between 2J 
and 2| inches. 

Depth of the eye of cross-head = 18-37 inches, or very nearly 
18| inches. 

Diameter of the journal of cross-head = 5*5 inches, or 5|- inches. 

Length of journal of cross-head = 6*19 inches, or very nearly 6l 
inches. 

Thickness of the web of cross-head at middle = 4-6 inches, or 
somewhat more than 4|^ inches. 

Breadth of web of cross-head at middle = 17-15 inches, or 
between VJ-^-q and 17| inches. 

Thickness of web of cross-head at journal = 3-93 inches, or 
very nearly 4 inches. 

Breadth of web of cross-head at journal = 6*46 inches, or nearly 
6J inches. 

Diameter of piston rod = 6-4 inches, or Q'l inches. 

Length of part of piston rod in piston = 12-8 inches, or 12| 
inches. 

Major diameter of part of piston rod in cross-head = 06-8 inches, 
or nearly Q^ inches. 

Minor diameter of part of piston rod in cross-head = 5*76 inches, 
or 5f inches. 

Major diameter of part of piston rod in piston = 8-96 inches, 
or nearly 9 inches. 

Minor diameter of part of piston rod in piston = 7-36 inches, 
or between 7^ and 7^^ inches. 

Depth of gibs and cutter through cross-head = 6-72 inches, or 
very nearly 6| inches. 

Thickness of gibs and cutter through cross-head = 1-35 inches, 
or between IJ and IJ inches. 



THE STEAM ENGINE. 139 

Depth of cutter through piston = 5*45 inches, or nearly 5J inches. 

Thickness of cutter through piston =^ 2-24 inches, or nearly 2^ 
inches. 

Diameter of connecting rod at ends = 6-08 inches, or nearly 
6|^o inches. 

Major diameter of part of connecting rod in cross-tail = 6*27 
inches, or about 6^ inches. 

Minor diameter of part of connecting rod in cross-tail = 5'76 
inches, or nearly 5f inches. 

Breadth of butt = 9*98 inches, or very nearly 10 inches. 

Thickness of butt = 8 inches. 

Mean thickness of strap at cutter = 2*75 inches, or 2f inches. 

Mean thickness of strap above cutter = 2*06 inches, or some- 
\Yhat more than 2 inches. 

Distance of cutter from end of strap = 3*08 inches, or very 
nearly 3jL inches. 

Breadth of gibs and cutter through cross-tail = 6-73 inches, or 
very nearly 6f inches. 

Breadth of gibs and cutter through butt = 7*04 inches, or some- 
what more than 7 inches. 

Thickness of gibs and cutter through butt = 1*84 inches, or 
between If and 2 inches. 

These results are calculated from the following rules, which give 
correct results for all sizes of engines. 

Rule 1. To find the diameter of crank-pin journal. — Multiply 
the diameter of the cylinder in inches by -142. The result is the 
diameter of crank-pin journal in inches. 

BuLE 2. To find the length of cranJc-jnn journal. — Multiply the 
diameter of the cylinder in inches by '16. The product is the 
length of the crank-pin journal in inches. 

Rule 3. To find the breadth of the eye of cross-head. — Multiply 
the diameter of the cylinder in inches by -041. The product is 
the breadth of the eye in inches. 

BuLE 4. To find the depth of the eye of cross-head. — Multiply 
the diameter of the cylinder in inches by '286. The product is 
the depth of the eye of cross-head in inches. 

BuLE 5. To find the diameter of the journal of cross-head. — 
Multiply the diameter of the cylinder in inches by '086. The pro- 
duct is the diameter of the journal in inches. 

Rule 6. To find the length of the journal of cross-head. — Mul- 
tiply the diameter of the cylinder in inches by "097. The product 
is the length of the journal in inches. 

Bule 7. To find the thickness of the weh of cross-head at middle. 
— Multiply the diameter of the cylinder in inches by •072. The 
product is the thickness of the web of cross-head at middle in 
inches. 

BtULE 8. To find the breadth of web of cross-head at middle. — 
Multiply the diameter of the cylinder in inches by -268. The 
product is the breadth of the web of cross-head at middle in inches. 



140 THE PRACTICAL MODEL CALCULATOR. 

Rule 9. To find the tliichiess of the lueh of cross-head at journal. 
— Multiply the diameter of the cylinder in inches by '061. The 
product is the thickness of the web of cross-head at journal in 
inches. 

Rule 10. To find the breadth of iveh of cross-head at journal. — 
Multiply the diameter of the cylinder in inches by '101. The 
product is the breadth of the web of cross-head at journal in inches. 

Rule 11. To find the diameter of the piston rod. — Divide the 
diameter of the cylinder in inches by 10. The quotient is the 
diameter of the piston rod in inches. 

Rule 12. To find the length of the part of the piston rod in the 
ptiston. — Divide the diameter of the cylinder in inches by 5. The 
quotient is the length of the part of the piston rod in the piston in 
inches. 

Rule 13. To find the major diameter of the 'part of piston rod in 
cross-head. — Multiply the diameter of the cylinder in inches by 
•095. The product is the major diameter of the part of piston rod 
in cross-head in inches. 

Rule 14. To find the minor diameter of the part of piston rod in 
cross-head. — Multiply the diameter of the cylinder in inches by '09. 
The product is the minor diameter of the part of piston rod in 
cross-head in inches. 

Rule 15. To find the major diameter-of the part of piston rod in 
piston. — Multiply the diameter of the cylinder in inches by '14. 
The product is the major diameter of the part of piston rod in 
piston in inches. 

Rule 16. To find the minor diameter of the part of piston rod in 
piston. — Multiply the diameter of the cylinder in inches by -115. 
The product is the minor diameter of the part of piston rod in 
piston. 

Rule 17. To find the depth of gibs and cutter through cross- 
head. — Multiply the diameter of the cylinder in inches by '105. 
The product is the depth of the gibs and cutter through cross- 
head. 

Rule 18. To find the thickness of the gibs and cutter through 
cross-head. — Multiply the diameter of the cylinder in inches by 
•021. The product is the thickness of the gibs and cutter through 
cross-head. 

Rule 19. To find the depth of cutter through piston. — Multiply 
the diameter of the cylinder in inches by -085. The product is the 
depth of the cutter through piston in inches. 

Rule 20. To find the thickness of cutter through piston. — Mul- 
tiply the diameter of the cylinder in inches by -035. The product 
is the thickness of cutter through piston in inches. 

Rule 21. To find the diameter of connecting rod at ends. — Mul- 
tiply the diameter of the cylinder in inches by -095. The product 
is the diameter of the connecting rod at ends in inches. 

Rule 22. To find tlie major diameter of the part of connecting 
rod in cross-tail. — Multiply the diameter of the cylinder in inches 



THE STEAM ENGINE. 141 

by -098. The product is the major diameter of the part of con- 
nectiDg rod in cross-tail. 

Rule 23. To find the minor diameter of the part of connecting 
rod in cross-tail. — Multiply the diameter of the cylinder in inches 
by -09. The product is the minor diameter of the part of con- 
necting rod in cross-tail in inches. 

Rule 24. To find the hreadth of hutt. — Multiply the diameter 
of the cylinder in inches by 456. The product is the breadth of 
the butt in inches. 

Rule 25. To find the thickness of the hutt. — Divide the diameter 
of the cylinder in inches by 8. The quotient is the thickness of 
the butt in inches. 

Rule 26. To find the mean thickness of the strap at cutter. — 
Multiply the diameter of the cylinder in inches by -043. The pro- 
duct is the mean thickness of the strap at cutter. 

Rule 27. To find the mean thickness of the strap above cutter. — 
Multiply the diameter of the cylinder in inches by -032. The 
product is the mean thickness of the strap above cutter. 

Rule 28. To find the distance of cutter from end of strap. — 
Multiply the diameter of the cylinder in inches by -048. The 
product is the distance of cutter from end of strap in inches. 

Rule 29. To find the hreadth of the gibs and cutter through 
cross-tail. — Multiply the diameter of the cylinder in inches by 
•105. The product is the breadth of the gibs and cutter through 
cross-tail. 

Rule 30. To find the hreadth of the gibs and cutter through 
hutt. — Multiply the diameter of the cylinder in inches by 41. 
The product is the breadth of the gibs and cutter through butt in 
inches. 

Rule 31. To find the thickness of the gibs and cutter through 
hutt. — Multiply the diameter of the cylinder in inches by '029. 
The product is the thickness of the gibs and cutter through butt 
in inches. 

To find other parts of the engine which do not depend upon the 
stroke. Suppose it were required to find the thickness of the small 
eye of crank for an engine the diameter of whose cylinder is 64 
inches. According to the rule, the proper thickness of the small 
eye of crank is 4*04 inches. Again, suppose it were required to 
find the length of the small eye of crank. Hence, according to 
the rule, the proper length of the small eye of crank is 11'94 inches. 
Again, supposing it were required to find the proper thickness of the 
web of crank at pin centre ; that is to say, the thickness w^hich it 
would have if continued to the pin centre. According to the rule, 
the proper thickness for the web of crank at pin centre is 7;04 inches. 
Again, suppose it were required to find the breadth of the web of 
crank at pin centre ; that is to say, the breadth which it would 
have if it were continued to the pin centre. Hence, according to 
the rule, the proper breadth of the web of crank at pin centre is 
10-24 inches. 



142 THE PRACTICAL MODEL CALCULATOR. 

These results are calculated from the follo'^ing rules, which give 
the proper dimensions for engines of all sizes : 

Rule 1. To find the breadth of the small eye of crank. — Multiply 
the diameter of the cylinder in inches by -063. The product is 
the proper breadth of the small eye of crank in inches. 

Rule 2. To find the length of the small eye of crank. — Multiply 
the diameter of the cylinder in inches by -187. The product is 
the proper length of the small eye of crank in inches. 

Rule 3. To find the thickness of the iveb of crank at pin centre. — 
Multiply the diameter of the cylinder in inches by -11. The pro- 
duct is the proper thickness of the "web of crank at pin centre in 
inches. 

Rule 4. To find the breadth of the web of cranh at pin centre. — 
Multiply the diameter of the cylinder in inches by '16. The pro- 
duct is the proper breadth of crank at pin centre in inches. 

To illustrate the use of the succeeding rules, let us take the par- 
ticular example of an engine of 8 feet stroke and 64-inch cylinder, 
and let us suppose that the length of the connecting rod is 12 
feet, and the side rod 10 feet. We find by a previous rule that 
the diameter of the connecting rod at ends is 6-08, and the ratio 
between the diameters at middle and ends of a connecting rod, 
whose length is 12 feet, is 1-504. Hence, the proper diameter at 
middle of the connecting rod = 6*08 X 1-504 inches = 9-144 
inches. And again, we find the diameter of cylinder side rods at 
ends, for the particular engine which we have selected, is 4-10, and 
the ratio between the diameters at middle and ends of cylinder 
side rods, whose lengths are 10 feet, is 1-42. Hence, according to 
the rules, the proper diameter of the cylinder side rods at middle 
is equal to 4-1 X 1-42 inches = 5-82 inches. 

To find some of those parts of the engine which do not depend 
upon the stroke. Suppose we take the particular example of an 
engine the diameter of whose cylinder is 64 inches. We find from 
the following rules that 

Diameter of cylinder side rods at ends = 4*1 inches, or 4^^ 
inches. 

Breadth of butt = 4-93 inches, or very nearly 5 inches. 

Thickness of butt = 3-9 inches, or 3^ inches. 

Mean thickness of strap at cutter = 2-06 inches, or a little more 
than 2 inches. 

Mean thickness of strap below cutter = 1-47 inches, or very 
nearly IJ inches. 

Depths of gibs and cutter = 5-12 inches, or a little more than 
5Jjj inches. 

Thickness of gibs and cutter = 1-03 inches, or a little more than 
1 inch. 

Diameter of main centre journal = 11-71 inches, or very nearly 
llf incnes. 

Length of main centre journal = 17*6 inches, or 17| inches. 



THE STEAM ENGINE. 143 

Depth of eye round end studs of lever = 4*75 inches, or 4f mches. 

Thickness of eye round end studs of lever = 3-33 inches, or 3-J 
inches. 

Diameter of end studs of lever = 4-48 inches, or very nearly 4J 
inches. 

Length of end studs of lever = 4-86 inches, or between 4f and 
5 inches. 

Diameter of air-pump studs = 2*91 inches, or nearly 3 inches. 

Length of air-pump studs = 3-16 inches, or nearly 3i inches. 

These results were obtained from the following rules, which will 
be found to give the proper dimensions for all sizes of engines. 

Rule 1. To find tJie diameter of cylinder side rods at ends. — 
Multiply the diameter of the cylinder in inches by -065. The 
product is the diameter of the cylinder side rods at ends in inches. 

Rule 2. To find the hreadth of butt in inches. — Multiply the 
diameter of the cylinder in inches by 'OTT. The product is the 
breadth of the butt in inches. 

Rule 3. To find the thickness of the butt. — Multiply the diameter 
of the cylinder in inches by '061. The product is the thickness of 
the biitt in inches. 

Rule 4. To find the mean thickness of strap at cutter. — Mul- 
tiply the diameter of the cylinder in inches by -032. The product 
is the mean thickness of the strap at cutter. 

Rule 5. To find the mean thickness of strap beloio cutter. — Mul- 
tiply the diameter of the cylinder in inches by -023. The product 
is the mean thickness of strap below cutter in inches. 

Rule 6. To find the depth of gibs and cutter. — Multiply the 
diameter of the cylinder in inches by -08. The product is the 
depth of the gibs and cutter in inches. 

Rule 7. To find the thickness of gibs and cutter. — Multiply the 
diameter of the cylinder in inches by '016. The product is the 
thickness of gibs and cutter in inches. 

Rule 8. To find the diameter of the main centre journal. — Mul- 
tiply the diameter of the cylinder in inches by -183. The product 
is the diameter of the main centre journal in inches. 

Rule 9. Tofi.nd the length of the main centre journal. — Multiply 
the diameter of the cylinder in inches by -275. The product is 
the diameter of the cylinder in inches. 

Rule 10. To find the depth of eye round end studs of lever. — 
Multiply the diameter of the cylinder in inches by '074. The pro- 
duct is the depth of the eye round end studs of lever in inches. 

Rule 11. To find the thickness of eye round end studs of lever. 
— Multiply the diameter of the cylinder in inches by -052. The 
product is the thickness of eye round end studs of lever in inches. 

Rule 12. To find the diameter of the end studs of lever. — Mul- 
tiply the diameter of the cylinder in inches by '07. The product 
is the diameter of the end studs of lever in inches. 

Rule 13. To find the letigth of the end studs of lever. — Multiply 



144 THE PRACTICAL MODEL CALCULATOR. 

tlie diameter of the cylinder in inches by "076. The product is the 
length of the end studs of lever in inches. 

Rule 14. To find the diameter of the air-pump studs. — Multiply 
the diameter of the cylinder in inches by '045. The product is 
the diameter of the air-pump studs in inches. 

Rule 15. To find the length of the air-pump studs. — Multiply 
the diameter of the cylinder in inches by -049. The product is the 
length of the air-pump studs in inches. 

The next rule gives the proper depth in inches across the centre 
of the side lever, when, as is generally the case, the side lever is 
of cast iron. It will be observed that the depth is made to depend 
upon the diameter of the cylinder and the length of the lever, and 
not at all upon the length of the stroke, except indeed in so far as 
the length of the lever may depend upon the length of the stroke. 
Suppose it were required to find the proper depth across the centre 
of a side lever whose length is 20 feet, and the diameter of the 
cylinder 64 inches. According to the rule, the proper depth 
across the centre would be 39-26 inches. 

The following rule will give the proper dimensions for any size 
of engine : 

Rule. — To find the depth across the centre of the side lever. — 
Multiply the length of the side lever in feet by -7423 ; extract the 
cube root of the product, and reserve the result for a multiplier. 
Then square the diameter of the cylinder in inches ; extract the 
cube root of the result. The product of the final result and the 
reserved multiplier is the depth of the side lever in inches across 
the centre. 

Thus, to apply this rule to the particular example which we have 
selected, we have 

20 = length of side lever in feet. 
•7423 = constant multiplier. 

14-846 

and ^ 14-846 = 2-458 nearly. 

64 = diameter of cylinder in inches. 
64 ^ 



4096 



and ^ 4096 = 16 

Hence depth at centre = 16 X 2-458 inches = 39-33 inches, or 
between 39J and 39J inches. 



The next set of rules give the dimensions of several of the parts 
of the air-pump machinery which depend upon the diameter of the 
cylinder only. To illustrate the use of these rules, let us take the 
particular example of an engine the diameter of whose cylinder is 
64 inches. We find from the succeeding rules successively. 

Diameter of air-pump = 38-4 inches, or 38f inches. 



THE STEAM ENGINE. 145 

Thickness of the eye of air-pump cross-head = 1*58 inches, or 
a little more than 1^ inches. 

Depth of eye of air-pump cross-head = ll'Ol, or about 11 inches. 

Diameter of end journals of air-pump cross-head = 3*29 inches, 
or somewhat more than 3J inches. 

Length of end journals of air-pump cross-head = 3*7 inches, or 
S^Q inches. 

Thickness of the web of air-pump cross-head at middle = 2*76 
inches, or a little more than 2| inches. 

Depth of web of air-pump cross-head at middle = 10*29 inches, 
or somewhat more than 10|^ inches. 

Thickness of w^eb of air-pump cross-head at journal = 2*35 
inches, or about 2| inches. 

Depth of web of air-pump cross-head at journal = 3*89 inches, 
or about 3| inches. 

Diameter of air-pump piston rod when made of copper = 4*27 
inches, or about 4J inches. 

Depth of gibs and cutter through air-pump cross-head = 4*04 
inches, or a little more than 4 inches. 

Thickness of gibs and cutter through air-pump cross-head = '81 
inches, or about | inch. 

Depth of cutter through piston = 3 -27 inches, or somewhat 
more than 3J inches. 

Thickness of cutter through piston = 1*34 inches, or about 1| 
inches. 

These results were obtained from the following rules, and give 
the proper dimensions for all sizes of engines : 

Rule 1. To find the diameter of the air-pump. — Multiply the- 
diameter of the cylinder in inches by '6. The product is the 
diameter of air-pump in inches. 

Rule 2. To find the thickness of the eye of air-pump cross-head. 
— Multiply the diameter of the cylinder in inches by '025. The 
product is the thickness of the eye of air-pump cross-head in inches. 

Rule 3. To find the depth of eye of air-pump cross-head. — Mul- 
tiply the diameter of the cylinder in inches by -171. The product 
is the depth of the eye of air-pump cross-head in inches. 

Rule 4. To find the diameter of the journals of air-pump cross- 
head. — Multiply the diameter of the cylinder in inches by 'OSl. 
The product is the diameter of the end journals. 

Rule 5. To find the length of the end journals for air-pump 
cross-head. — Multiply the diameter of the cylinder in inches by 
•058. The product is the length of the air-pump cross-head jour- 
nals in inches. 

Rule 6. To find the thickness of the weh of air-pump> cross-head 
at middle. — Multiply the diameter of the cylinder in inches by -043. 
The product is the thickness at middle of the web of air-pump 
cross-head in inches. 

Rule 7. To find the depth at middle of the weh of air-pump cross- 
head. — Multiply the diameter of the cylinder in inches by '161. 
N 10 



146 THE PRACTICAL MODEL CALCULATOR. 

The product is the depth at middle of air-pump cross-head in 
inches. 

EuLE 8. To find the thickness of the weh of air-pump cross- 
head at journals. — Multiply the diameter of the cylinder in inches 
by -037. The product is the thickness of the web of air-pump 
cross-head at journals in inches. 

Rule 9. To find the depth of the air-pump cross-head weh at 
journals. — Multiply the diameter of the cylinder in inches by -061. 
The product is the depth at journals of the web of air-pump cross- 
head. 

Rule 10. To find the diameter of the air-pump piston rod when 
of copper. — Multiply the diameter of the cylinder in inches by 
•067. The product is the diameter of the air-pump piston rod, 
when of copper, in inches. 

Rule 11. To find the depth of gibs and cutter through air-pump 
cross-head. — Multiply the diameter of the cylinder in inches by 
•063. The product is the depth of the gibs and cutter through air- 
pump cross-head in inches. 

Rule 12. To find the thickness of the gihs and cutter through 
air-pump cross-head. — Multiply the diameter of the cylinder in 
inches by -013. The product is the thickness of the gibs and 
cutter in inches. 

Rule 13. To find the depth of cutter through piston. — Multiply 
the diameter of the cylinder in inches by '051. The product is the 
depth of the cutter through piston in inches. 

Rule 14. To find the thickness of cutter through air-pump 
piston. — Multiply the diameter of the cylinder in inches by •021. 
The product is the thickness of the cutter through air-pump piston. 



The next seven rules give the dimensions of the remaining parta 
of the engine which do not depend upon the stroke. To exemplify 
their use, suppose it were required to find the corresponding dimen- 
sions for an engine the diameter of whose cylinder is 64 inches. 
According to the rule, the proper diameter of the air-pump side 
rod would be 2-48 inches. Hence, according to the rule, the 
proper breadth of butt is 2*95 inches. According to the rule, the 
proper thickness of butt is 2-35 inches. According to the rule, 
the mean thickness of strap at cutter ought to be 1*24 inches. 
Hence, according to the rule, the mean thickness of strap below 
cutter is *91 inch. According to the rule, the proper depth for 
the gibs and cutter is 2*94 inches. According to the rule, the 
proper thickness of the gibs and cutter is ^63 inches. 

The following rules give the correct dimensions for all sizes of 
engines : 

Rule 1. To find the diameter of air-pump side rod at ends. — 
Multiply the diameter of the cylinder in inches by '039. The 
product is the diameter of the air-pump side rod at ends in inches. 

Rule 2. To find the breadth of butt for air-pump. — Multiply the 



THE STEAM ENGINE. 147 

diameter of the cylinder in inches by -046. The product is the 
breadth of butt in inches. 

KuLE 3. To find the thickness of hutt for air-pump. — Multiply 
the diameter of the cylinder in inches by -037. The product is 
the thickness of butt for air-pump in inches. 

Rule 4. To find the mean thickness of strap at cutter. — Multiply 
the diameter of the cylinder in inches by -019. The product is 
the mean thickness of strap at cutter for air-pump in inches. 

Rule 5. To find the mean thickness of strap below cutter. — Mul- 
tiply the diameter of the cylinder in inches by 0*14. The product 
is the mean thickness of strap below cutter in inches. 

Rule 6. To find the depth of gibs and cutter for air-pump. — 
Multiply the diameter of the cylinder in inches by 0-48. The 
product is the depth of gibs and cutter for air-pump in inches. 

Rule 7. To find the thickness of gibs and cutter for air-pump. — 
Divide the diameter of the cylinder in inches by 100. The 
quotient is the proper thickness of the gibs and cutter for air-pump 
in inches. 

With regard to other dimensions made to depend upon the 
nominal horse power of the engine : — Suppose that we take the 
particular example of an engine whose stroke is 8 feet, and dia- 
meter of cylinder 64 inches. We find that the nominal horse 
power of this engine is nearly 175. Hence we have successively, 

Diameter of valve shaft at journal in inches = 4*85, or between 
4f and 5 inches. 

Diameter of parallel motion shaft at journal in inches = 3-91, or 
very nearly 4 inches. 

Diameter of valve rod in inches = 2*44, or about 2f inches. 

Diameter of radius rod at smallest part in inches = 1-97, or 
very nearly 2 inches. 

Area of eccentric rod, at smallest part, in square inches = 8*37, 
or about 8f square inches. 

Sectional area of eccentric hoop in square inches = 8*75, or 8f 
square inches. 

Diameter of eccentric pin in inches = 2-24, or 2J inches. 

Breadth of valve lever for eccentric pin at eye in inches = 5*7, 
or very nearly 5| inches. 

Thickness of valve lever for eccentric pin at eye in inches = 3. 

Breadth of parallel motion crank at eye = 4*2 inches, or very 
nearly 4|^ inches. 

Thickness of parallel motion crank at eye = 1*76 inches, or 
about If inches. 

To find the area in square inches of each steam port. Suppose 
it were required to find the area of each steam port for an engine 
whose stroke is 8 feet, and diameter of cylinder 64 inches. Accord- 
ing to the rule, the area of each steam port would be 202*26 square 
inches. 

With regard to the rule, we may remark that the area of the 



148 THE PRACTICAL MODEL CALCULATOR. 

steam port ought to depend principally upon tlie cubical content of 
the cylinder, which again depends entirely upon the product of the 
square of the diameter of the cylinder and the length of the stroke 
of th'^ engine. It is well known, however, that the quantity of 
steam admitted by a small hole does not bear so great a proportion 
to the quantity admitted by a larger one, as the area of the one does 
to the area of the other ; and a certain allowance ought to be made 
for this. In the absence of correct theoretical information on this 
point, we have attempted to make a proper allowance by supplying 
a constant ; but of course this plan ought only to be regarded as 
an approximation. Our rule is as follows : 

Rule. — To find the area of each steam port. — Multiply the 
square of the diameter of the cylinder in inches by the length of 
the stroke in feet ; multiply this product by 11 ; divide the last 
product by 1800 ; and, finally, to the quotient add 8. The result 
is the area of each steam port in square inches. 

To show the use of this rule, we shall apply it to a particular 
example. We shall apply it to an engine whose stroke is 6 feet, 
and diameter of cylinder 30 inches. Then, according to the rule, 
we have 

30 = diameter of the cylinder in inches. 
_80 ^ 

900 = square of diameter. 
6 = length of stroke in feet. 

5400 
11 



59400 -V- 1800 = 33 

8 = constant to be added. 

41 = ^rea of steam port in square inches. 

When the length of the opening of steam port is from any cir- 
cumstance found, the corresponding depth in inches may be found, 
by dividing the number corresponding to the particular engine, by 
the given length in inches : conversely, the length may be found, 
when for some reason or other the depth is fixed, by dividing the 
number corresponding to the particular engine, by the given depth 
in inches : the quotient is the length in inches. 

The next rule is useful for determining the diameter of the steam 
pipe branching ofi* to any particular engine. Suppose it were 
required to find the diameter of the branch steam pipe for an 
engine whose stroke is 8 feet, and diameter of cylinder 64 inches. 
According to the rule, the proper diameter of the steam pipe 
would be 13*16 inches. 

The following rule will be found to give the proper diameter of 
steam pipe for all sizes of engines. 

Rule. — To find the diameter of branch steam pipe. — Multiply 
together the square of the diameter of the cylinder in inches, the 



THE STEAM ENGINE. 149 

length of the stroke in feet, and '00498; to the product add 10-2, 
and extract the square root of the sum. The result is the diameter 
of the steam pipe in inches. 

To exemplify the use of this rule we shall take an engine whose 
stroke is 8 feet, and diameter of cylinder 64 inches. In this case 
we have as follows : — 

64 = diameter of cylinder in inches. 



4096 = square of diameter. 
8 = length of stroke in feet. 



32768 

•00498 = constant multiplier. 

163-18 
10-2 = constant to be added. 

173-38 
and >/ 173-38 = 13-16. 

To find the diameter of the pipes connected with the engine. 
They are made to depend upon the nominal horse power of the 
engine. Suppose it were required to apply this rule to determine 
the size of the pipes for two marine engines, whose strokes are 
each 8 feet, and diameters of cylinder each 64 inches. We find 
the nominal horse power of each of these engines to be 174-3. 
Hence, according to the rules, we have in succession. 

Diameter of waste water pipe = 15-87 inches, or between 15f 

and 16 inches. 
Area of foot-valve passage = 323 square inches. 
Area of injection pipe = 14*88 square inches. 
If the injection pipe be cylindrical, then by referring to the 
table of areas of circles, we see that its diameter would be about 
4f inches. 

Diameter of feed pipe = 4-12 inches, or between 4 and 4|- 

inches. 
Diameter of waste steam pipe = 12-17 inches, or nearly 12 J 

inches. 
Diameter of safety valve, 

When one is used =14-05 inches. 
When two are used = 9-94 inches. 
When three are used = 8-12 inches. 
When four are used = 7*04 inches. 

These results were obtained from the following rules, which will 
give the correct dimensions for all sizes of engines. 

Rule 1. To find the diameter of waste water pipe. — Multiply 
the square root of the nominal horse power of the engine by 1-2. 
The product is the diameter of the waste water pipe in inches. 

KuLE 2. To find the area of foot-valve passage. — Multiply the 
n2 



150 THE PRACTICAL MODEL CALCULATOR. 

nominal horse power of the engine by 9 ; divide the product by 5 ; 
add 8 to the quotient. The sum is the area of foot-valve passage 
in square inches. 

Rule 3. To find the area of injection pipe. — Multiply the nomi- 
nal horse power of the engine by -069 ; to the product add 2*81. 
The sum is the area of the injection pipe in square inches. 

Rule 4. To find the diameter of feed pipe. — Multiply the nomi- 
nal horse power of the engine by -04 ; to the product add 3 ; extract 
the square root of the sum. The result is the diameter of the feed 
pipe in inches. 

Rule 5. To find the diameter of waste steam pipe. — Multiply 
the collective nominal horse power of the engines by -375 ; to the 
product add 16*875 ; extract the square root of the sum. The 
final result is the diameter of the waste steam pipe in inches. 

Rule 6. To find the diameter of the safety valve when only one 
is used. — To one-half the collective nominal horse power of the 
engines add 22*5 ; extract the square root of the sum. The result 
is the diameter of the safety valve when only one is used. 

Rule 7. To find the diameter of the safety valve when two are 
used. — Multiply the collective nominal horse power of the engines 
by '25 ; to the product add 11'25 ; extract the square root of the 
sum. The result is the diameter of the safety valve when two 
are used. 

Rule 8. To find the diameter of the safety valve when three are 
used. — To one-sixth of the collective nominal horse power of the 
engines add 7*5 ; extract the square root of the sum. The result 
is the diameter of the safety valve where three are used. 

Rule 9. To find the diameter of the safety valve when four are 
used. — Multiply the collective nominal horse power of the engines 
by '125 ; to the product add 5*625 ; extract the square root of the 
sum. The result is the diameter of the safety valve when four 
are used. 

Another rule for safety valves, and a preferable one for low 
pressures, is to allow "8 of a circular inch of area per nominal 
horse power. 

The next rule is for determining the depth across the web of the 
main beam of a land engine. Suppose we wished to find the proper 
depth at the centre of the main beam of a land engine whose main 
beam is 16 feet long, and diameter of cylinder 64 inches. Accord- 
ing to the rule, the proper depth of the web across the centre is 
46*17 inches. This rule gives correct dimensions for all sizes of 
engines. 

Rule. — To find the depth of the tveh at the centre of the main 
beam of a land engine. — Multiply together the square of the di- 
ameter of the cylinder in inches, half the length of the main beam 
in feet, and the number 3 ; extract the cube root of the product. 
The result is the proper depth of the web of the main beam across 
the centre in inches, when the main beam is constructed of cast 
iron. 



THE STEAM ENGINE. 151 

To illustrate this rule we shall take the particular example of an 
engine whose main beam is 20 feet long, and the diameter of the 
cylinder 64 inches. In this case we have 

6-4 = diameter of cylinder in inches. 
64 





4096 = square of the diameter. 






10 = i 


length of main beam in 


feet. 




40960 








3 = constant multiplier. 






122880 



122880(49-714 = 







= ^122880 


4 


16 


64 




4 


16 


58880 




4 


32 


53649 




8 


4800 


5231 




4 


1161 


5112 




120 


5961 


119 




9 


1242 


74 




129 


7203 


35 




9 


10 






138 


730 






9 


10 







147 741 

To find the depth of the main beam across the ends. Suppose 
it were required to find the depth at ends of a cast-iron main beam 
whose length is 20 feet, when the diameter of the cylinder is 64 
inches. The proper depth will be 19*89 inches. 

The following rule gives the proper dimensions for all sizes of 
engines. 

Rule. — To find the depth of main beam at ends. — Multiply to- 
gether the square of the diameter of the cylinder in inches, half 
the length of the main beam in feet, and the number -192 ; extract 
the cube root of the product. The result is the depth in inches of 
the main beam at ends, when of cast iron. 

To illustrate this rule, let us apply it to the particular example 
of an engine whose main beam is 20 feet long, and the diameter 
of the cylinder 64 inches. In this case we have as follows : 
64 = diameter of cylinder in inches. 

4096 = square of diameter of cylinder. 
10 = J length of main beam in feet. 



40960 
•192 = constant multiplier. 

'7864-32 



152 THE PRACTICAL MODEL CALCULATOR. 









7864-32 (19-89 = 


= -^7864-32 


1 


1 


1 




1 


1 


6864 




1 


2 


5859 




2 


300 


1005 




1 


351 


898 




30 


651 


lOT 




9 


432 






39 


1083 






9 


4 






48 


112 






9 


4 







57 116 

so that, according to tlie rule, the depth at ends is nearly 20 inches. 

To find the dimensions of the feed-pump in cubic inches. Sup- 
pose "sve take the particular example of an engine whose stroke is 
8 feet, and diameter of cylinder 64 inches. The proper content of 
the feed-pump would be 1093-36 cubic inches. Suppose, now, 
that the cold-water pump was suspended from the main beam at a 
fourth of the distance between the centre and the end, so that its 
stroke would be 2 feet, or 24 inches. In this case the area of the 
pump would be equal to 1093*36 -f- 24 = 45-556 square inches ; 
so that we conclude that the diameter is between 7| and 7f inches. 
Conversely, suppose that it was wished to find the stroke of the 
pump when the diameter was 5 inches. We find the area of the 
pump to be 19-635 square inches ; so that the stroke of the feed- 
pump must be equal to 1093-36 -r- 19-635 = 55*69 inches, or very 
nearly 55f inches. 

This rule will be found to give correct dimensions for all sizes 
of engines : 

Rule. — To find the content of the feed-pump. — Multiply the 
square of the diameter of the cylinder in inches by the length of 
the stroke in feet ; divide the product by 30. The quotient is the 
content of the feed-pump in cubic inches. 

Thus, for an engine whose stroke is 6 feet, and diameter of cylin- 
der 50 inches, we have, 

50 = diameter of cylinder. 
50 ^ 



2500 = square of the diameter of the cylinder. 
6 = length of stroke in feet. 



30 )15000 

500 = content of feed-pump in cubic inches. 
To determine the content of the cold-water pump in cubic feet. 
To illustrate this, suppose we take the particular example of an en- 



THE STEAM ENGINE. 163 

gine whose stroke is 8 feet, and diameter of cylinder 64 inches. 
Suppose, now, the stroke of the pump to he 5 feet, then the area 
equal to 7*45 -^ 5 = 1'49 square feet = 214*56 square inches ; 
we see that the diameter of the pump is about 16J inches. Again, 
suppose that the diameter of the cold-water pump was 20 inches, 
and that it was required to find the length of its stroke. The area 
of the pump is 314-16 square inches, or 314-16 -i- 144 = 2-18 
square feet; so that the stroke of the pump is equal to 7 '45 -?- 
2-18 = 3-42 feet. 

The content is calculated from the following rule, which will he 
found to give correct dimensions for all sizes of engines : 

Rule. — To find the content of the cold-ivater pump. — Multiply 
the square of the diameter of the cylinder in inches by the length 
of the stroke in feet ; divide the product by 4400. The quotient 
is the content of the cold-water pump in cubic feet. 

To explain this rule, we shall take the particular example of an 
engine whose stroke is 5J feet, and diameter of cylinder 60 inches. 
In this case we have in succession, 

60 = diameter of cylinder in inches. 
60 ^ 



3600 = square of the diameter of cylinder. 
5J = length of stroke in feet. 

4400) 19800 

4-5 = content of cold water pump in cubic feet. 

To determine the proper thickness of the large eye of crank for 
fly-wheel shaft when the crank is of cast iron. The crank is some- 
times cast on the shaft, and of course the thickness of the large 
eye is not then so great as when the crank is only keyed on the 
shaft, or rather there is then no large eye at all. To illustrate the 
use of this rule, we shall apply it to the particular example of an 
engine whose stroke is 8 feet, and diameter of cylinder 64 inches. 
Hence, according to the rule, the proper thickness of the large eye 
of crank when of cast iron is 8*07 inches. For a marine engine 
of 8 feet stroke and 64 inch cylinder, the thickness of the large 
eye of crank is about 5| inches. The difference is thus about 2J 
inches, which is an allowance for the inferiority of cast iron to 
malleable iron. 

The following rule will be found to give correct dimensions for 
all sizes of engines : 

Rule. — To find the thickness of the large eye of crank for fly- 
wheel shaft when of cast iron. — Multiply the square of the length 
of the crank in inches by 1*561, and then multiply the square of the 
diameter of the cylinder in inches by -1235 ; multiply the sum of 
these products by the square of the diameter of cylinder in inches ; 
divide this product by 666-283; divide this quotient by the length 
of the crank in inches ; finally extract the cube root of the quotient. 



154 THE PRACTICAL MODEL CALCULATOR. 

The result is the proper thickness of the large eye of crank for 
fly-wheel shaft in inches, when of cast iron. 

As this rule is rather complicated, we shall show its application 
to the particular example already selected. 

48 = length of crank in inches. 

48 



2304 = square of length of crank in inches. 
1-561 = constant multiplier. 

3596^ 

64 = diameter of cylinder in inches. 
64 ^ 

4096 = square of the diameter of cylinder. 
•1235 = constant multiplier. 

505-8 
3596-5 

4102-3 = sum of products. 
4096 = square of the diameter of cylinder. 

666-283 )16803020- 8 

length of crank=48 ) 25219-045 

525 -397 
and -^525-397 = 8-07 nearly. 

To find the breadth of the web of crank at the centre of the fly- 
wheel shaft, that is to say, the breadth which it would have if it 
were continued to the centre of the fly-wheel shaft. Suppose it 
were required to find the breadth of the crank at the centre of the 
fly-wheel shaft for an engine whose stroke is 8 feet, and diameter 
of cylinder 64 inches. According to the rule, the proper breadth 
is 22-49 inches. According to a former rule, the breadth of the 
web of a cast iron crank of an engine whose stroke is 8 feet, and 
diameter of cylinder 64 inches, is about 18 inches. The difi'erence 
between these two is about 4|- inches ; which is not too great an 
allowance for the inferiority of the cast iron. 

The following rule will be found to give correct dimensions for 
all sizes of engines : 

Rule. — To find the breadth of the iveh of crank at fly -wheel shaft, 
when of cast iron. — Multiply the square of the length of the crank 
in inches by 1-561, and then multiply the square of the diameter 
of the cylinder in inches by -1235 ; multiply the square root of the 
sum of these products by the square of the diameter of the cylinder 
in inches ; divide the product by 23-04, and finally extract the 
cube root of the quotient. The final result is the breadth of the 
crank at the centre of the fly-wheel shaft, when the crank is of 
cast iron. 

As this rule is rather complicated, we shall illustrate it by show- 



THE STEAM ENGINE. 155 

ing its application to the particular example of an engine whose 
stroke is 8 feet, and diameter of cylinder 64 inches. 

64 = diameter of cylinder in inches. 
64 



4096 = square of the diameter of cylinder. 
•1235 = constant multiplier. 

505-8 

48 = length of crank in inches. 
48 



2304 = square of the length of crank. 
1*561 = constant multiplier. 

3596-5 
505-8 

4102-3 = sum of products, 
v/ 4102-3 = 64-05 nearly. 

4096 = square of the diameter of 
constant divisor = 23-04 )262348-5 [cylinder. 

1138 6-66 nearl y. 
and ^11386-66 = 22-49. 

To determine the thickness of the web of crank at the centre of 
the fly-wheel shaft ; that is to say, the thickness which it would 
have if it were continued so far. Suppose it were required to find 
the thickness of web of crank at the centre of fly-wheel shaft of 
an engine whose stroke is 8 feet, and diameter of cylinder 64 
inches. According to the rule, the proper thickness would be 
11-26 inches. The proper thickness of web at centre of paddle 
shaft for a marine engine whose stroke is 8 feet, and diameter of 
cylinder 64 inches, is nearly 9 inches. The difl'erence between the 
two thicknesses is about 2^ inches, which is not too great an allow- 
ance for the inferiority of cast iron to malleable iron. 

The following rule will be found to give correct dimensions for 
all sizes of engines : 

Rule. — To find tlie tJiichness of the weh of crank at centre of 
jiy-ivheel shafts when of cast iron. — Multiply the square of the 
length of the crank in inches by 1-561, and then multiply the 
square of the diameter of the cylinder in inches by -1235 ; multi- 
ply the square root of the sum of these products by the square of 
the diameter of the cylinder in inches ; divide this product by 
184-32 ; finally extract the cube root of the quotient. The result 
is the thickness of the web of crank at the centre of the fly-wheel 
shaft when of cast iron, in inches. 

As this rule is rather complicated, we shall illustrate it by apply- 
ing it to the particular engine which we have already selected. 



156 THE PKACTICAL MODEL CALCULATOR. 

48 = length of crank in inches. 

48 



2304 = square of length of crank. 
1-561 = constant multiplier. 

3596^ 

64 = diameter of cylinder in inches. 
64 



4096 = square of the diameter of cylinder. 
•1235 = constant multiplier. 



505-8 
3596-5 



4102-3 = sum of products. 



and v/ 4102-3 = 64-05 nearly. 

4096 = square of diameter. 
Constant divisor = 184-32 )262348^ 

1423-33 

and ^ 1423-33 = 11-24 

To find the proper diameter of the fly-wheel shaft at its smallest 
part, when, as is usually the case, it is of cast iron. Suppose it 
were required to find the diameter of the fly-wheel shaft for an 
engine whose stroke is 8 feet, and diameter of cylinder 64 inches. 
According to the rule, the diameter would be 17*59 inches. It is 
obvious enough that the fly-wheel shaft stands in much the same 
relation to the land engine, as the paddle shaft does to the marine 
engine. According to a former rule, the diameter of the paddle 
shaft journal of a marine engine whose stroke is 8 feet, and dia- 
meter of cylinder 64 inches, is about 14 inches. The difi"erence 
betwixt the diameter of the paddle shaft for the marine engine, 
and the diameter of the fly-wheel shaft for the corresponding land 
engine is about 3J inches. This will be found to be a very proper 
allowance for the different circumstances connected with the land 
engine. 

The following rule will be found to give correct dimensions for all 
sizes of engines. 

Rule. — To find the diameter of the fly-wheel shaft at smallest 
part, when it is of east iron. — Multiply the square of the diameter 
of the cylinder in inches by the length of the crank in inches ; 
extract the cube root of the product ; finally multiply the result 
by -3025. The result is the diameter of the fly-wheel shaft at 
smallest part in inches. 

We shall illustrate this rule by applying it to the particular 
engine which we have already selected. 



THE STEAM ENGINE. 157 





64 = 
64 


■ diameter of cylinder in inclies. 




4096 = 
48 = 


■ square of the diameter. 

■ length of crank in inches. 




196608 


25 





5 


196608 (58-15 = ^ 196608 
125 


5 
5 

10 

5 


25 
50 

7500 
1264 


71608 
70112 

1496 
1011 


150 

8 


8764 
1328 


485 


158 
8 

166 

8 


10092 

2 

1011 

2 





174 1013 

and 58-15 x -3025 = 1759 
which agrees "with the number given by a former rule. 

To determine the sectional area of the fly-wheel rim when of 
cast iron. Suppose it were required to find the sectional area of 
the rim of a fly-wheel for an engine whose stroke is 8 feet, and 
diameter of cylinder 64 inches, the diameter of the fly-wheel itself 
being 30 feet. According to the rule, the sectional area of the 
rim in square inches = 146*4 X -813 = 119*02. We may remark 
that this calculation has been made on the supposition that the fly- 
wheel is so connected with the engine, as to make exactly one revo- 
lution for each double stroke of the piston. If the fly-wheel is so 
connected with the engine as to make more than one revolution for 
each double stroke, then the rim does not need to be so heavy as 
we make it. If, on the contrary, the fly-wheel does not make a 
complete revolution for each double stroke of the engine, then it 
ought to be heavier than this rule makes it. 

Rule. — To find the sectional area of the rim of the fiy-wheel 
when of cast iron. — Multiply together the square of the diameter 
of the cylinder in inches, the square of the length of the stroke 
in feet, the cube root of the length of the stroke in feet, and 6*125 ; 
divide the final product by the cube of the diameter of the fly-wheel 
in feet. The quotient is the sectional area of the rim of fly-wheel 
in square inches, provided it is of cast iron. 

As this rule is rather complicated, we shall endeavour to illustrate 
it by showing its application to a particular engine. We shall 
apply the rule to determine the sectional area of the rim of fly- 



158 THE PRACTICAL MODEL CALCULATOR. 

wheel for an engine whose stroke is 8 feet, diameter of cylinder 50 
inches ; the diameter of the fly-wheel being 20 feet. For this 
engine we have as follows : 

2500 = square of diameter of cylinder. 
64 = square of the length of stroke. 



160000 

2 = cube root of the length of stroke. 



320000 
6*125 = constant multiplier. 

1960000 

therefore sectional area in square inches = 1960000 -r- 20^ = 
1960000 -- 8000 = 1960 -r- 8 = 245. 

In the following formulas we denote the diameter of the cylinder 
in inches by D, the length of the crank in inches by R, the length 
of the stroke in feet, and the nominal horse power of the engine 
by H.P. 

MARINE ENGINES. DIMENSIONS OP SEVERAL OF THE PARTS OF THE 

SIDE LEVER. 

Depth of eye round end studs of lever = '074 X D. 
Thickness of eye round end studs of lever = -052 X D. 
Diameter of end studs, in inches = -07 X D. 
Length of end studs, in inches = -076 X D. 
Diameter of air-pump studs, in inches = -045 X D. 
Length of air-pump studs, in inches = -049 X D. 

Depth of cast iron side lever across centre, in inches = D^ X 

{•7423 X length of lever in feet}*. 

MARINE ENGINE. — DIMENSIONS OF SEVERAL PARTS OF AIR-PUMP 
CROSS-HEAD. 

Diameter of air-pump, in inches = '6 X D. 
Thickness of eye for air-pump rod, in inches = "025 X D. 
Depth of eye for air-pump rod, in inches = '171 X D. 
Diameter of end journals, in inches = -051 X D. 
Length of end journals, in inches = -058 X D. 
Thickness of web at middle, in inches = '043 x D. 
Depth of web at middle, in inches = '161 X D. 
Thickness of web at journal = -037 X D. 
Depth of web at journal = -061 X D. 

MARINE ENGINE. — DIMENSIONS OF THE PARTS OF AIR-PUMP 
PISTON-ROD. 

Diameter of air-pump piston-rod, when of copper, in inches = 
•067 X D. 

Depth of gibs and cutter through cross-head, in inches = 
•063 X D. 



THE STEAM ENGINE. ISST 

Thickness of gibs and cutter through cross-head, in inches = 
•013 X D. 

Depth of cutter through piston, in inches = "051 X D. 
Thickness of cutter through piston, in inches = '021 x B. 

MARINE ENGINE. — DIMENSIONS OF THE REMAINING PARTS OF THE 
AIR-PUMP MACHINERY. 

Diameter of air-pump side rods at ends, in inches = '039 X D. 

Breadth of butt, in inches = -046 x D. 

Thickness of butt, in inches = -037 X D. 

Mean thickness of strap at cutter, in inches = '019 X D. 

Mean thickness of strap below cutter, in inches = '014 X D. 

Depth of gibs and cutter, in inches = '048 X D. 

Thickness of gibs and cutter in inches = D -i- 100. 

MARINE AND LAND ENGINES. — AREA OF STEAM PORTS. 

Area of each steam port, in square inches = 11 X ^ X D^ -4- 

1800 -f 8. 

MARINE AND LAND ENGINES. — DIMENSIONS OF BRANCH STEAM PIPES. 



Diameter of each branch steam pipe = \/ -00498 xlxD^X 10-2. 

MARINE ENGINE. — DIMENSIONS OF SEVERAL OF THE PIPES CONNECTED 
WITH THE ENGINE. 



Diameter of waste water pipe, in inches = 1*2 X ^/ H.P. 
Area of foot-valve passage, in square inches = 1*8 X H.P.+ 8. 
Area of injection pipe, in square inches = -069 X H.P. -f- 2*81. 
Diameter of feed pipe, in inches = v/ -04 x H.P. -f 3. 
Diameter of waste steam pipe in inches =\/-375xH.P.-j-16'875. 

MARINE AND LAND ENGINES. — DIMENSIONS OF SAFETY-VALVES. 



Diam. of safety-valve, when one only is used =^/•5xH.P.-(-22•5. 
Diam. of safety-valve, when two are used = -v/-25 x H.P. -f 11*25. 
Diam. of safety-valve, when three are used = \/ -167 x H.P. +7 '5. 
Diam. of safety-valve, when four are used = \/ -125 X H.P. -f 5*625. 

LAND ENGINE. — DIMENSIONS OF MAIN BEAM. 

Depth of web of main beam across centre = 

■^ 3 X D^ X half length of main beam in feet. 
Depth of main beam at ends = 

■^ -192 X D^ X half length of main beam, in feet. 

LAND AND MARINE ENGINES. — CONTENT OF FEED-PUMP. 

Content of feed-pump, in cubic inches = D^ x ? -^ 30. 

LAND ENGINES. — CONTENT OF COLD WATER PU3IP. 

Content of cold water pump, in cubic feet = D^ x ? h- 4400. 



160 THE PRACTICAL MODEL CALCULATOR. 

LAND ENGINES. — DIMENSIONS OF CRANK. 

Thickness of large eye of crank, in inches = 

■^J)^ X (1-561 X R2 + -1235 D^) -^ (R x 666-283). 
Breadth of web of crank at fly-wheel shaft centre, in inches = 

^B^ X s/ (1-561 X R^ + -1235 x J)') -f- 23-04. 
Thickness of web of crank at fly-wheel shaft centre, in inches = 
^ D^ x >/ (1-561 X R^ + -1235 x D^) -f- 184-32. 

LAND ENGINES. — DIMENSIONS OF FLY-WHEEL SHAFT. 



Diameter of fly-wheel shaft, when of cast iron = 3025 X ^RxD^ 



DIMENSIONS OF PARTS OF LOCOMOTIVES. 

DIAMETER OF CYLINDER. 

In locomotive engines, the diameter of the cylinder varies less 
than either the land or the marine engine. In few of the locomotive 
engines at present in use is the diameter of the cylinder greater 
than 16 inches, or less than 12 inches. The length of the stroke of 
nearly all the locomotive engines at present in use is 18 inches, and 
there are always two cylinders, which are generally connected to 
cranks upon the axle, standing at right angles with one another. 

AREA or INDUCTION PORTS. 

Rule. — To find the size of the steam ports for the locomotive 
engine. — Multiply the square of the diameter of the cylinder by 
•068. The product is the proper size of the steam ports in square 
inches. 

Required the proper size of the steam ports of a locomotive 
engine whose diameter is 15 inches. Here, according to the rule, 
size of steam ports = -068 x 15 x 15 = -068 X 225 = 15-3 square 
inches, or between 15^ and 15|- square inches. 

After having determined the area of the ports, we may easily 
find the depth when the length is given, or, conversely, the length 
when the depth is given. Thus, suppose we knew that the length 
was 8 inches, then we find that the depth should be 15-3 -r- 8 = 
1-9125 inches, or nearly 2 inches; or suppose we knew the depth 
was 2 inches, then we would find that the length was 15-3 -i- 2 = 
7-65 inches, or nearly Tf inches. 

AREA or EDUCTION PORTS. 

The proper area for the eduction ports may be found from the fol- 
lowing rule. 

Rule. — To find the area of the eduction ports. — Multiply the 
square of the diameter of the cylinder in inches by -128. The 
product is the area of the eduction ports in square inches. 

Required the area of the eduction ports of a locomotive engine, 



THE STEAM ENGINE. 161 

when the diameter of the cylinders is 13 inches. In this example 
we have, according to the rule, 

Area of eduction port = -128 X 13^ = -128 x 169 = 21-632 
inches, or between 21J and 21| square inches. 

BREADTH OF BRIDGE BETWEEN PORTS. 

The breadth of the bridges between the eduction port and the 
induction ports is usually between | inch and 1 inch. 

DIAMETER OF BOILER. 

It is obvious that the diameter of the boiler may vary very con- 
siderably ; but it is limited chiefly by considerations of strength ; 
and 3 feet are found a convenient diameter. Rules for the strength 
of boilers will be given hereafter. 

Rule. — To find the inside diameter of the 5oz7er. ^Multiply the 
diameter of the cylinder in inches by 341. The product is the 
inside diameter of the boiler in inches. 

Required the inside diameter of the boiler for a locomotive 
engine, the diameter of the cylinders being 15 inches. 

In this example we have, according to the rule, 

Inside diameter of boiler = 15 X 3*11 = 46-65 inches, 
or about 3 feet lOf inches. 

LENGTH OF BOILER. 

The length of the boiler is usually in practice between 8 feet and 
81 feet. 

DIAMETER OF STEAM DOME, INSIDE. 

It is obvious that the diameter of the steam dome may be varied 
considerably, according to circumstances ; but the first indication 
is to make it large enough. It is usual, however, in practice, to 
proportion the diameter of the steam dome to the diameter of the 
cylinder; and there appears to be no great objection to this. The 
following rule will be found to give the diameter of the dome 
usually adopted in practice. 

Rule. — To find the diameter of the steam dome. — Multiply the 
diameter of the cylinder in inches by 1-43. The product is the 
diameter of the dome in inches. 

Required the diameter of the steam dome for a locomotive engine 
whose diameter of cylinders is 13 inches. In this example we 
have, according to the rule. 

Diameter of steam dome = 1-43 X 13 = 18-59 inches, 
or about 18J inches. 

HEIGHT OF STEAM DOME. 

The height of the steam dome may vary. Judging from prac- 
tice, it appears that a uniform height of 2J feet would answer 
very well. 

o2 11 



162 THE PRACTICAL MODEL CALCULATOR. 

DIAMETER OF SAFETY-VALVE. 

In practice the diameter of tlie safety-valve varies considerably. 
The following rule gives the diameter of the safety-valve usually 
adopted in practice. 

Rule. — To find the diameter of the safety-valve. — Divide the 
diameter of the cylinder in inches by 4. The quotient is the dia- 
meter of the safety-valve in inches. 

Required the diameter of the safety-valves for the boiler of a 
locomotive engine, the diameter of the cylinder being 13 inches. 
Here, according to the rule, diameter of safety-valve = 13 -r- 4 = 3J 
inches. A larger size, however, is preferable, as being less likely 
to stick. 

DIAMETER OF VALVE SPINDLE. 

The following rule will be found to give the correct diameter of 
the valve spindle. It is entirely founded on practice. 

Rule. — To find the diameter of the valve spindle. — Multiply the 
diameter of the cylinder in inches by '076. The product is the 
proper diameter of the valve spindle. 

Required the diameter of the valve spindle for a locomotive 
engine whose cylinders' diameters are 13 inches. 

In this example we have, according to the rule, diameter of valve 
spindle = 13 x -076 = -988 inches, or very nearly 1 inch. 

DIAMETER OF CHIMNEY. 

It is usual in practice to make the diameter of the chimney equal 
to the diameter of the cylinder. Thus a locomotive engine whose 
cylinders' diameters are 15 inches would have the inside diameter 
of the chimney also 15 inches, or thereabouts. This rule has, at 
least, the merit of simplicity. 

AREA OF FIRE-GRATE. 

The following rule determines the area of the fire-grate usually 
given in practice. We may remark, that the area of the fire-grate 
in practice follows a more certain rule than any other part of the 
engine appears to do ; but it is in all cases much too small, and 
occasions a great loss of power by the urging of the blast it renders 
necessary, and a rapid deterioration of the furnace plates from 
excessive heat. There is no good reason why the furnace should 
not be nearly as long as the boiler : it would then resemble the 
furnace of a marine boiler, and be as manageable. 

Rule. — To find the area of the fire-grate. — Multiply the diameter 
of the cylinder in inches by -77. The product is the area of the fire- 
grate in superficial feet. 

Required the area of the fire-grate of a locomotive engine, the 
diameters of the cylinders being 15 inches. 

In this example we have, according to the rule, 

Area of fire-grate = -77 X 15 = 11*55 square feet, 
or about llj square feet. Though this rule, however, represents 



THE STEAM ENGINE. 163 

the usual practice, the area of the fire-grate should not be contingent 
upon the size of the cylinder, but upon the quantity of steam to be 
raised. 

AREA OF HEATING SURFACE, 

In the construction of a locomotive engine, one great object is to 
obtain a boiler which will produce a sufficient quantity of steam with 
as little bulk and weight as possible. This object is admirably ac- 
complished in the construction of the boiler of the locomotive en- 
gine. This little barrel of tubes generates more steam in an hour 
than was formerly raised from a boiler and fire occupying a con- 
siderable house. This favourable result is obtained simply by ex- 
posing the water to a greater amount of heating surface. 

In the usual construction of the locomotive boiler, it is obvious 
that we can only consider four of the six faces of the inside fire-box 
as effective heating surface ; viz. the crown of the box, and the 
three perpendicular sides. The circumferences of the tubes are also 
effective heating surface ; so that the whole efi"ective heating sur- 
face of a locomotive boiler may be considered to be the four faces 
of the inside fire-box, plus the sura of the surfaces of the tubes. 
Understanding this to be the effective heating surface, the following 
rule determines the average amount of heating surface usually given 
in practice. 

Rule. — To find the effective heating surface. — Multiply the square 
of the diameter of the cylinder in inches by 5 ; divide the product 
by 2. The quotient is the area of the effective heating surface in 
square feet. 

Required the effective heating surface of the boiler of a locomotive 
engine, the diameters of the cylinders being 15 inches. 

In this example we have, according to the rule, 

Effective heating surface = 15^ X 5 ^ 2 = 225 X 5 ^ 2 = 1125 -r- 
2 = 562|- square feet. 

According to the rule which we have given for the fire-grate, the 
area of the fire-grate for this boiler would be about 11|^ square feet. 
We may suppose, therefore, the area of the crown of the box to be 
^12 square feet. The area of the three perpendicular sides of the 
inside fire-box is usually three times the area of the crown ; so that 
the effective heating surface of the fire-box is 48 square feet. Hence 
the heating surface of the tubes = 526*5 — - 48 = 478*5 square feet. 
The inside diameters of the tubes are generally about If inches ; 
and therefore the circumference of a section of these tubes, ac- 
cording to the table, is 5*4978 inches. Hence, supposing the 
tube to be 8J feet long, the surface of one = 5*4978 x 8|^ -r- 12 = 
•45815 X 8|- = 3*8943 square feet; and, therefore, the number of 
tubes = 478*5 -=-3*8943 = 123 nearly. The amount of heating sur- 
face, however, like that of grate surface, is properly a function of 
the quantity of steam to be raised, and the proportions of both, 
given hereafter, will be found to answer well for boilers of every 
description. 



164 THE PRACTICAL MODEL CALCULATOR. 

AHEA or WATER-LEVEL. 

This, of coui'se, varies with the different circumstances of the 
boiler. The average area may be found from the following rule. 

Rule. — To find the area of the ivater-level. — Multiply the diame- 
ter of the cylinder in inches by 2-08. The product is the area of 
the water-level in square feet. 

Required the area of the water-level for a locomotive engine, 
whose cylinders' diameters are 14 inches. 

In this case we have, according to the rule, 

Area of water-level = 14 X 2-08 = 29*12 square feet. 

CUBICAL CONTEXT OE WATER IN BOILER. 

This, of course, varies not only in different boilers, but also in 
the same boiler at different times. The following rule is supposed 
to give the average quantity of water in the boiler. 

Rule. — To find the cubical content of tJie water in the boiler, — - 
Multiply the square of the diameter of the cylinder in inches by 9 : 
divide the product by 40. The quotient is the cubical content of 
the water in the boiler in cubic feet. 

Required the average cubical content of the water in the boiler 
of a locomotive engine, the diameters of the cylinders being 14 
inches. In this example we have, according to the rule. 

Cubical content of water = 9 X 14^ -h 40 = 44-1 cubic feet. 

CONTENT or PEED-PUMP. 

In the locomotive engine, the feed-pump is generally attached to 
the cross-head, and consequently it has the same stroke as the pis- 
ton. As we have mentioned before, the stroke of the locomotive 
engine is generally in practice 18 inches. Hence, assuming the 
stroke of the feed-pump to be constantly 18 inches, it only remains 
for us to determine the diameter of the ram. It may be found from 
the following rule. 

Rule. — To find the diameter of the feed-puni}:) ram. — Multiply 
the square of the diameter of the cylinder in inches by 'Oil. The 
product is the diameter of the ram in inches. 

Required the diameter of the ram for the feed-pump for a loco- 
motive engine whose diameter of cylinder is 14 inches. In this 
example we have, according to the rule. 

Diameter of ram = -Oil x 14^ = -Oil X 196 = 2156 inches, 
or between 2 and 2^ inches. 

CUBICAL CONTENT OF STEAM ROOM. 

The quantity of steam in the boiler varies not only for different 
boilers, but even for the same boiler in different circumstances. 
But when the locomotive is in motion, there is usually a certain 
proportion of the boiler filled with the steam. Including the dome 
and the steam pipe, the content of the steam room will be found 
usually to be somewhat less than the cubical content of the water. 



THE STEAM ENGINE. 165 

But as it is desirable that it should be increased, we give the fol- 
lowing rule. 

KuLE. — To find the cubical content of the steam room. — Multiply 
the square of the diameter of the cylinder in inches by 9 ; divide 
the product by 40. The quotient is the cubical content of the 
steam room in cubic feet. 

Required the cubical content of the steam room in a locomotive 
boiler, the diameters of the cylinders being 12 inches. 

In this example we have, according to the rule, 

Cubical content of steam room = 9 X 12^ -r- 40 = 9 X 144 ^ 40 = 
32-4 cubic feet. 

CUBICAL CONTENT OF INSIDE riRE-BOX ABOVE FIRE-BARS. 

The following rule determines the cubical content of fire-box 
usually given in practice. 

Rule. — To find the cubical content of inside fire-box above fire- 
bars. — Divide the square of the diameter of the cylinder in inches 
by 4. The quotient is the content of the inside fire-box above fire- 
bars in cubic feet. 

Required the content of inside fire-box above fire-bars in a loco- 
motive engine, when the diameters of the cylinders are each 15 
inches. 

In this example we have, according to the rule, 

Content of inside fire-box above fire-bars = 15^ -r-4 = 225 -i- 4 = 
56J cubic feet. 

THICKNESS OF THE PLATES OF BOILER. 

In general, the thickness of the plates of the locomotive boiler is 
I inch. In some cases, however, the thickness is only ^ inch. 

INSIDE DIAMETER OF STEAM PIPE. 

The diameter usually given to the steam pipe of the locomotive 
engine may be found from the following rule. 

Rule. — To find the diameter of the steam pipe of the locomotive 
engine. — Multiply the square of the diameter of the cylinder in 
inches by :03. The product is the diameter of the steam pipe in 
inches. 

Required the diameter of the steam pipe of a locomotive engine, 
the diameter of the cylinder being 13 inches. Here, according to 
the rule, diameter of steam pipe = -03 X 13^ = -03 X 169 = 5-07 
inches ; or a very little more than 5 inches. The steam pipe is 
usually made too small in engines intended for high speeds. 

DIAMETER OF BRANCH STEAM PIPES. 

The following rule gives the usual diameter of the branch steam 
pipe for locomotive engines. 

Rule. — To find the diameter of the branch steam pipe for the lo- 
comotive engine. — Multiply the square of the diameter of the cylin- 
der in inches by -021. The product is the diameter of the branch, 
steam pipe for the locomotive engine in inches. 



166 THE PRACTICAL MODEL CALCULATOR. 

Required the diameter of tlie brancli steam pipes for a locomo- 
tive engine, when the cylinder's diameter is 15 inches. Here, ac- 
cording to the rule, diameter of branch pipe = -021 x 15^ = -021 X 
225 = 4-725 inches, or about 4| inches. 

DIAMETER OF TOP OF BLAST PIPE. 

The diameter of the top of the blast pipe may be found from the 
following rule. 

Rule. — To find tlie diameter of the top of the blast pipe. — Mul- 
tiply the square of the diameter of the cylinder in inches by 0-lT. 
The product is the diameter of the top of the blast pipe in inches. 

The diameter of a locomotive engine is 13 inches ; required the 
diameter of the blast pipe at top. Here, according to the rule, 
diameter of blast pipe at top = -017 X 13^ = -017 X 169 =2-873 
inches, or between 2| and 3 inches ; but the orifice of the blast 
pipe should always be made as large as the demands of the blast 
will permit. 

DIAMETER OF FEED PIPES. 

There appear to be no theoretical considerations which would 
lead us to determine exactly the proper size of the feed pipes. 
Judging from practice, however, the following rule will be found to 
give the proper dimensions. 

Rule. — To find the diameter of the feed p)ipe§. — Multiply the 
diameter of the cylinder in inches by -141. The product is the 
proper diameter of the feed pipes. 

Required the diameter of the feed pipes for a locomotive engine, 
the diameter of the cylinder being 15 inches. 

In this example we have, according to the rule. 

Diameter of feed-pipe = 15 X -141 = 2-115 inches, 
or between 2 and 2\ inches. 

DIAMETER OF PISTON ROD. 

The diameter of the piston rod for the locomotive engine is 
usually about one-seventh the diameter of the cylinder. Making 
practice our guide, therefore, we have the following rule. 

Rule. — To find the diameter of the piston rod for the locomotive 
e7igine. — Divide the diameter of the cylinder in inches by 7. The 
quotient is the diameter of the piston rod in inches. 

The diameter of the cylinder of a locomotive engine is 15 inches ; 
required the diameter of the piston rod. Here, according to the 
rule, diameter of piston rod =15 -f- 7 = 2^ inches. 

THICKNESS OF PISTON. 

The thickness of the piston in locomotive engines is usually about 
two-sevenths of the diameter of the cylinder. Making practice our 
guide, therefore, we have the following rule. 

Rule. — To find the thickness of the piston in the locomotive en- 
gine. — Multiply the diameter of the cylinder in inches by 2 ; divide 



THE STEAM ENGINE. ^ 167 

the product by 7. The quotient is the thickness of the piston in 
inches. 

The diameter of the cylinder of a locomotive engine is 14 inches ; 
required the thickness of the piston. Here, according to the rule, 
thickness of piston = 2x14-^-7=4 inches. 

DIAMETER OF CONNECTING RODS AT MIDDLE. 

The following rule gives the diameter of the connecting rod at 
middle. The rule, we may remark, is entirely founded on practice. 

KuLE. — To find the diameter of the connecting rod at middle of 
the locomotive engine. — Multiply the diameter of the cylinder in 
inches by '21. The product is the diameter of the connecting rod 
at middle in inches. 

Required the diameter of the connecting rods at middle for a 
locomotive engine, the diameter of the cylinders being twelve 
inches. 

For this example we have, according to the rule. 

Diameter of connecting rods at middle = 12 X '21 = 2*52 inches, 
or 2} inches. 

DIAMETER OF BALL ON CROSS-HEAD SPINDLE. 

The diameter of the ball on the cross-head spindle may be found 
from the following rule. 

Rule. — To find the diameter of the ballon cross-head spindle of 
a locomotive engine. — Multiply the diameter of the cylinder in 
inches by '23. The product is the diameter of the ball on the 
cross-head spindle. 

Required the diameter of the ball on the cross-head spindle of a 
locomotive engine, when the diameter of the cylinder is 15 inches. 
Here, according to the rule. 

Diameter of ball = -23 X 15 = 3-45 inches, or nearly 3J inches. 

DIAMETER OF THE INSIDE BEARINGS OF THE CRANK AXLE, 

It is obvious that the inside bearings of the crank axle of the 
locomotive engine correspond to the paddle-shaft journal of the 
marine engine, and to the fly-wheel shaft journal of the land-engine. 
We may conclude, therefore, that the proper diameter of these bear- 
ings ought to depend jointly upon the length of the stroke and the 
diameter of the cylinder. In the locomotive engine the stroke is 
usually 18 inches, so that we may consider that the diameter of the 
bearing depends solely upon the diameter of the cylinder. The 
following rule will give the diameter of the inside bearing. 

Rule. — To find the diameter of the inside hearing for the loco- 
motive engine. — Extract the cube root of the square of the diameter 
of the cylinder in inches ; multiply the result by -96. The product 
is the proper diameter of the inside bearing of the crank axle for the 
locomotive engine. 

Required the diameter of the inside bearing of the crank axle 



168 THE PKACTICAL MODEL CALCULATOR. 

for a locomotive engine whose cylinders are of IB-inch diameters. 
In this example we have, according to the rule, 



13 


= diameter of 


cylinder in inches. 


13 








169 


= square 


of the diameter of cylinder. 










169(5-5289 = ^169 


5 


25 




125 


5 


25 




44000 


5 


50 




41875 


10 


7500 




2625 


5 


7T5 




1820 


150 


8275 




805 


5 


800 




726 


155 


9075 




79 


5 


3 






160 


910 






5 


3 







165 913 

and diameter of bearing = 5*5289 X -96 = 5*31 inches nearly; or 
between 5J and 5J inches. 

DIAMETER OF THE OUTSIDE BEARINGS OF THE CRANK AXLE. 

The crank axle, in addition to resting upon the inside bearings, 
is sometimes also made to rest partly upon outside bearings. 
These outside bearings are added only for the sake of steadiness, 
and they do not need to be so strong as the inside bearings. The 
proper size of the diameter of these bearings may be found from 
the following rule. 

Rule. — To find the diameter of outside hearings for the locomo- 
tive engine. — Multiply the square of the diameters of the cylinders 
in inches by '396 ; extract the cube root of the product. The result 
is the diameter of the outside bearings in inches. 

Required the proper diameter of the outside bearings for a loco- 
motive engine, the diameter of its cylinders being 15 inches. 

In this example we have, according to the rule, 

15 = diameter of cylinders in inches. 
_15 ^ 

225 = square of diameter of cylinder. 
•396 = constant multiplier. 

8"94 



THE STEAM ENGINE. 169 




4 



16 


89-1(4-466 = 
64 


^89-1 


4 


16 


25100 




4 


32 


21184 




8 


4800 


3916 




4 


496 


3528 




120 


5296 


388 




4 


512 


358 




124 


5808 






4 


8 






128 


588 






4 


8 






132 


596 







Hence diameter of outside bearing = 4*466 inches, or very 
nearly 4J inches. 

DIAMETER OF PLAIN PART OP CRANK AXLE. 

It is usual to make the plain part of crank axle of the same sec- 
tional area as the inside bearings. Hence, to determine the sec- 
tional area of the plain part when it is cylindrical, we have the fol- 
lowing rule. 

Rule. — To determine the diameter of the plain part of crank axle 
for the locomotive engine. — Extract the cube root of the square of 
the diameter of the cylinder in inches; multiply the result by -96. 
The product is the proper diameter of the plain part of the crank 
axle of the locomotive engine in inches. 

Required the diameter of the plain part of the crank axle for the 
locomotive engine, whose cylinders' diameters are 14 inches. Ii: 
this example we have, according to the rule, 

14 = diameter of cylinder in inches. 
14 
196 = square of the diameter of cylinder. 




5 


•0 
25 


196(5-808 = 
125 


^196 


5 
5 


25 

50 


71-000 
70-112 




10 

5 


7500 
1264 


•888 




150 

8 


8764 
1328 






158 
8 

166 
8 

174 


10092 







170 THE PRACTICAL MODEL CALCULATOR. 

Hence the plain part of crank axle = 5*808 X '96 = 5*58 nearly, 
or a little more than 5J inches. 

DIAMETER OF CRANK PIN. 

The following rule gives the proper diameter of the crank pin. It is 
obvious that the crank pin of the locomotive engine is not altogether 
analogous to the crank pin of the marine or land engine, and, like 
them, ought to depend upon the diameter of the cylinder, as it is 
usually formed out of the solid axle. 

Rule. — To find the diameter of the cranlc pin for the locomotive 
engine. — Multiply the diameter of the cylinder in inches by '404. 
The product is the diametor of the crank pin in inches. 

Required the diameter of the crank pin of a locomotive engine 
whose cylinders' diameters are 15 inches. 

In this example we have, according to the rule, 

Diameter of crank pin = 15 X -404 = 6*06 inches, or about 6 
inches. 

LENGTH OP CRANK PIN. 

The length of the crank pin usually given in practice may be 
found from the following rule. 

Rule. — To find the length of the crank pin. — Multiply the di- 
ameter of the cylinder in inches by '233. The product is the 
length of the crank pins in inches. 

Required the length of the crank pins for a locomotive engine 
with a diameter of cylinder of 13 inches. 

In this example we have, according to the rule, 

Length of crank pin = 13 X -233 = 3*029 inches, 
or about 3 inches. The part of the crank axle answering to the 
crank pin is usually rounded very much at the corners, both to give 
additional strength, and to prevent side play. 

These then are the chief dimensions of locomotive engines ac- 
cording to the practice most generally followed. The establish- 
ment of express trains and the general exigencies of steam locomo- 
tion are daily introducing innovations, the effect of which is to make 
the engines of greater size and power : but it cannot be said that a 
plan of locomotive engine has yet been contrived that is free from 
grave objections. The most material of these defects is the neces- 
sity that yet exists of expending a large proportion of the power in 
the production of a draft ; and this evil is traceable to the inade- 
quate area of the fire-grate, which makes an enormous rush of air 
through the fire necessary to accomplish the combustion of the fuel 
requisite for the production of the steam. To gain a sufficient area 
of fire-grate, an entirely new arrangement of engine must be 
adopted : the furnace must be greatly lengthened, and perhaps it 
may be found that short upright tubes, or the very ingenious ar- 
rangement of Mr. Dimpfell, of Philadelphia, may be introduced 
with advantage. Upright tubes have been found to be more 
effectual in raising steam than horizontal tubes ; but the tube 
plate in the case of upright tubes would be more liable to burn. 



THE STEAM ENGINE. 171 

We here give the preceding rules in formulas, in the belief that 
those well acquainted with algebraic symbols prefer to have a rule 
expressed as a formula, as they can thus see at once the different 
operations to be performed. In the following formulas we denote 
the diameter of the cylinder in inches by D. 

LOCOMOTIVE ENGINE. — PARTS OF THE CYLINDER. 

Area of induction ports, in square inches = '068 X D^. 
Area of eduction ports, in square inches = '128 X D^ 
Breadth of bridge betAveen ports between f inch and 1 inch. 

LOCOMOTIVE ENGINE. — PARTS OF BOILER. 

Diameter of boiler, in inches = 3-11 X D. 

Length of boiler between 8 feet and 12 feet. 

Diameter of steam dome, inside, in inches = 1*43 X D. 

Height of steam dome = 2|- feet. 

Diameter of safety valve, in inches = D -r- 4. 

Diameter of valve spindle, in inches = '076 X D. 

Diameter of chimney, in inches = D. 

Area of fire-grate, in square feet = '77 X D. 

Area of heating surface, in square feet = 5 X D^ -r- 2. 

Area of water level, in square feet = 2*08 X D. 

Cubical content of water in boiler, in cubic feet = 9 X D^ -j- 40. 

Diameter of feed-pump ram, in inches = "Oil X D^. 

Cubical content of steam room, in cubic feet = 9 X D^ -^ 40. 

Cubical content of inside fire-box above fire bars, in cubic feet = 

D^ -r- 4. 

Thickness of the plates of boiler = f inch. 

LOCOMOTIVE ENGINE. — DIMENSIONS OF SEVERAL PIPES. 

Inside diameter of steam pipe, in inches = '03 X D^. 
Inside diameter of branch steam pipe, in inches = '021 X D^ 
Inside diameter of the top of blast pipe = "017 X D^. 
Inside diameter of the feed pipes = '141 x D. 

LOCOMOTIVE ENGINE. — DIMENSIONS OF SEVERAL MOVING PARTS. 

Diameter of piston rod, in inches = D -i- 7. 

Thickness of piston, in inches = 2 D -r- 7. 

Diameter of connecting rods at middle, in inches = '21 X D. 

Diameter of the ball on cross-head spindle, in inches = -23 X D. 

Diameter of the inside bearings of the crank axle, in inches = 
•96 X ^^D\ 

Diameter of the plain part of crank axle, in inches = '96 X ^ J)\ 

Diameter of the outside bearings of the crank axle, in inches = 
^ -396 X D\ 

Diameter of crank pin, in inches = '404 X D. 

Length of crank pin, in inches = '233 X D. 



172 



THE PRACTICAL MODEL CALCULATOR. 



Table of the Pressure of Steam, in Inches of Mercury, at dif- 
ferent Temperatures. 



Tempe- 


















rature, 
Fahren- 
heit. 


Dalton. 


Ure. 


Young. 


Ivory. 


Tredgold. 


Southern. 


Eobison. 


Watt. 


0° 


0-08 


... 




... 


... 


... 


... 


... 


10 


0-12 


... 




... 


... 


... 


... 


... 


20 


0-17 




0-h 




... 






... 


32 


0-26 


0-20 


0-18 




0-17 


0-16 


0-00 


... 


40 


0-34 


0-25 


0-20 




0-24 


0-22 


0-10 


... 


50 


0-49 


0-36 


0-36 


0-'36 


0-37 


0-33 


0-20 


... 


60 


0-65 


0-52 


0-53 




0-55 


0-48 


0-35 




70 


0-87 


0-73 


0-75 


0-73 


0-78 


0-68 


0-55 


0-77 


80 


1-16 


1-01 


1-05 




1-11 


0-95 


0-82 


... 


90 


1-59 


1-36 


1-44 


i-36 


1-53 


1-34 


1-18 




100 


2-12 


1-86 


1-95 


... 


2-08 


1-84 


1-60 


1-55 


110 


2-79 


2-45 


2-62 


2-46 


2-79 


2-56 


2-25 




120 


3-63 


3-30 


3-46 


... 


3-68 


3-46 


3-00 




130 


4-71 


4-37 


4-54 


4-41 


4-81 


4-43 


3-95 


... 


140 


6-05 


5-78 


5-88 


... 


6-21 


5-75 


5-15 


5-14 


150 


7-73 


7-53 


7-55 


7-42 


7-94 


7-46 


6-72 


... 


160 


9-79 


9-60 


9-62 


... 


10-05 


9-52 


8-65 


8-92 


170 


12-31 


12-05 


12-14 


12-05 


12-60 


12-14 


11-05 


11-37 


180 


15-38 


15-16 


15-23 




15-67 


15-20 


14-05 


12-73 


190 


18-98 


19-00 


18-96 


18-93 


19-00 




17-85 


19-00 


200 


23-51 


23-60 


23-44 


... 


23-71 




22-65 




210 


28-82 


28-88 


28-81 


28-81 


28-86 




28-62 




212 


30-00 


30-00 


30-00 


30-00 


30-00 


30-00 


30-00 


29-40 


220 


35-18 


35-54 


35-19 




34-92 




35-8 


33-65 


230 


44-60 


43-10 


42-47 


42-63 


42-00 




44-5 


40 


240 


53-45 


51-70 


51-66 


... 


50-24 




54-9 


49-0 



Table of the Temperature of Steam at different Pressures in At- 
mospheres. 



Pressure in 
Atmospheres. 


French 
Academy. 


Dr. Ure. 


Young. 


Ivor 


'. Tredgold. 


Southern. 


Robison. 


Watt. 


Franklin 
Institute. 


1st At. 


212-0° 


212° 


212° 


211 


1° 212° 




212° 


212° 


212° 


2d At. 


250-5 


250-0 


240-3 


24£ 


250 


250-3 




252-5 


250-0 


3d At. 


275-2 


275-0 


271 


.. 


274 




267 




275-2 


4tli At. 


293-7 


291-5 


288 


29C 


294 


293-4 






291-5 


5th At. 


308-8 


304-5 


302 


.. 


309 


... 


.. 






304-5 


6th At. 


320-4 


315-5 






322 


... 


.. 






315-5 


7th At. 


331-7 


325-5 










., 






326-5 


8th At. 


342-0 


336-0 




337 


342 


343-6 


., 






336-0 


9th At. 


350-0 


345-0 










„ 






345-0 


10th At. 


358-9 


... 
















352-5 


nth At. 


366-8 


... 
















... 


12th At. 


374-0 




... 




372 








... 


... 


13th At. 


380-6 




















14th At. 


386-9 






,. 














15th At. 


392-8 






.. 












883-8 


16th At. 


398-5 




... 






... 






... 


... 


17th At. 


403-8 




... 




... 








... 




18th At. 


408-9 


... 


... 


.. 


... 


... 


.. 




... 


... 


19th At. 


413-9 


... 




.. 


... 


... 


.. 




... 


... 


20th At. 


418-5 








414 










405 


30th At. 


457-2 


... 






... 










... 


40th At. 


466-6 


... 






... 








... 


... 


50th At. 


510-6 


... 










•• 




... 


... 



THE STEAM ENGINE. 



173 



Table of the Expansion of Air hy Heat. 



Fahren. 




Fahren. 




Fahren. 




32 


.... 1000 


61 


.... 1069 


90 


.... 1132 


33 


.... 1002 


62 


.... 1071 


91 


.... 1134 


34 

35 


.... 1004 
.... 1007 


63 


.... 1073 


92 

93 


.... 1136 
.... 1138 


64 


.... 1075 


36 


.... 1009 


65 


.... 1077 


94 


.... 1140 


37 


.... 1012 


66 


.... 1030 


95 


.... 1142 


38 


.... 1015 


67 


.... 1080 


96 


.... 1144 


39 


.... 1018 


68 


.... 1034 


97 


.... 1146 


40 


.... 1021 


69 


.... 1087 


98 


.... 1148 


41 


.... 1023 


70 


.... 1089 


99 


.... 1150 


42 


.... 1025 


71 


.... 1091 


100 


.... 1162 


43 


.... 1027 


72 


.... 1093 


110 


.... 1173 


44 


.... 1030 


73 


.... 1095 


120 


.... 1194 


45 


.... 1032 


74 


.... 1097 


130 


.... 1215 


46 


.... 1034 


75 


.... 1099 


140 


.... 1235 


47 


.... 1036 


76 


.... 1101 


150 


.... 1255 


48 


.... 1038 


77 


.... 1104 


160 


.... 1275 


49 


.... 1040 


78 


.... 1106 


170 


.... 1295 


50 


.... 1043 


79 


.... 1108 


180 


.... 1315 


61 


.... 1045 


80 


.... 1110 


190 


.... 1334 


52 


.... 1047 


81 


.... 1112 


200 


.... 1364 


68 


.... 1050 


82 


.... 1114 


210 


.... 1372 


54 


.... 1052 


83 


.... 1116 


212 


.... 1376 


55 


.... 1055 


84 


.... 1118 


302 


.... 1558 


56 


.... 1057 


85 


.... 1121 


392 


.... 1739 


67 


.... 1059 


86 


.... 1123 


482 


.... 1919 


58 


.... 1062 


87 


.... 1125 


672 


.... 2098 


69 


.... 1064 


88 


.... 1128 


680 


.... 2312 


60 


.... 1066 


89 


.... 1130 







STRENGTH OF MATERIALS. 



The chief materials, of which it is necessary to record the strength 
in this place, are cast and malleable iron ; and many experiments 
have been made at different times upon each of these substances, 
though not with any very close correspondence. The following is 
a summary of them : — 



Materials. 


c 


s 


^ 


M 


i^o-.-'ito".":::::;:::;::::::: 

Malleable 


16300 \ 
36000 / 
60000 
80000 


8100 
9000 


69120000 
91440000 


6530000 
6770000 


Wire 





The first column of figures, marked C, contains the mean strength 
of cohesion on an inch section of the material ; the second, marked 
S, the constant for transverse strains ; the third, marked E, the 
constant for deflections ; and the fourth, marked M, the modulus 
of elasticity. The introduction of the hot blast iron brought with 
it the impression that it was less strong than that previously in use, 
and the experiments which had previously been confided in as 
giving results near enough the truth, for all practical purposes, 
were no longer considered to be applicable to the new state of 
things. New experiments were therefore made. The following 
Table gives, we have no doubt, results as nearly correct as can be 
required or attained ; — 
p2 



174 



THE PRACTICAL MODEL CALCULATOR. 



RESULTS OF EXPERIMENTS ON THE STRENGTH AND OTHER PRO- 
PERTIES OF CAST IRON. 

In the following Table each bar is reduced to exactly one inch 
square ; and the transverse strength, which may be taken as a 
criterion of the value of each Iron, is obtained from a mean between 
the experiments upon it; — first on bars 4 ft. 6 in. between the 
supports ; and next on those of half the length, or 2 ft. 3 in. be- 
tween the supports. All the other results are deduced from the 
4 ft. 6 in. bars. In all cases the weights were laid on the middle 
of the bar. 



Dickerson's, Newark, N. J 

Ponkey, No. 3. Cold Blast 

Devon, No. 3. Hot Blast* 

Oldberry, No. 3. Hot Blast 

Pattison, N. J. Hot Blast* 

Beaufort, No. 3. Hot Blast 

Pennsylvanian 

Bute, No. 1. Cold Blast 

Wind Mill End. No. 2. Cold Blast 

Old Park, No. 2. Cold Blast 

Beaufort, No. 2. Hot Blast 

Low Moor. No. 2. Cold Blast- • • • 

Bufifery, No. 1. Cold Blast* 

Brimbo, No. 2. Cold Blast 

Apedale, No. 2. Hot Blast 

Oldberry, No. 2. Cold Blast 

Pentwyn, No. 2 

Maesteg, No. 2 

Muirkirk, No. 1. Cold Blast* 

Adelphi, No. 2. Cold Blast 

Blania, No. 3. Cold Blast 

Devon, No. 3. Cold Blast* 

Gartsherrie, No. 3. Hot Blast • 

Frood, No. 2. Cold Blast 

Lane End, No. 2. 

Carron, No. 3. Cold Blast* 

Dundyvan. No. 3. Cold Blast 

Maesteg (Marked Red) 

Corbyns Hall, No. 2 

Pontypool, No. 2 

Wallbrook, No. 3 

Milton, No. 3. Hot Blast 

Bufifery, No. 1. Hot Blast* 

Level, No. 1. Hot Blast 

Pant, No. 2 

Level, No. 2. Hot Blast 

W. S. S., No. 2 

Eagle Foundry, No. 2. Hot Blast 

Elsiear, No. 2. Cold Blast 

Varteg, No. 2. Hot Blast 

Coltham, No. 1. Hot Blast 

Carroll. No. 2. Cold Blast 

Muirkirk, No. 1. Hot Blast* 

Bierley, No. 2 

Coed-Talon, No. 2. Hot Blast* ■ • 
Coed-Talon, No. 2. Cold Blast*- • 

Monkland, No. 2. Hot Blast 

Ley's Works, No. 1. Hot Blast- - 

Milton, No. 1. Hot Blast 

Plaskynaston, No. 2. Hot Blast - 



7-030 
7-122 

7-251 



7-066 
7-071 
7-049 
7-108 
7-0o5 
7-079 
7-017 
7-017 
7-059 
7-038 
7-038 
7-113 
7-080 
7-159 
7-285 
7-017 
7-031 
7-028 
7-094 
7-087 
7-038 
7-007 
7-080 
6-979 
7-051 
6-998 
7-080 
6-975 
7-031 
7-041 
7-038 
6-9-28 
7-007 
7-128 
7-069 
6-953 
7-185 
6-969 
6-955 
6-916 
6-957 
6-976 
6-916 



se = 



18470000 
17211000 
22473650 
22733400 
17873100 
16802000 
15379500 
15163000 
16490000 
14607000 
16301000 
14509500 
15381200 
14911666 
14852000 
14307500 
15193000 
13959500 
14003550 
1.3815500 
142S1466 
22907700 
13894000 
13112666 
15787666 
16246966 
16534000 
13971500 
13845866 
13136500 
15.394766 
15852-500 
13730500 
15452500 
15280900 
15241000 
14953333 
14211000 
12586500 
15012000 
15510066 
17036000 
13294400 
16156133 
14322500 
14304000 
12259500 
11539333 
11974500 
13341633 



%?, 


<SSx; 


« 


d§ 


« 


m 


1 


•s» 


l„-2 


i'ff. 


-^7 


1^ 




s=S 




-§.2 


Hi 
III 


m 

Hi 


1 
1 


ill 


510 


532 


600 


1-530 


567 


595 


581 


1-747 


537 





537 


1-09 


543 


517 


530 


1-005 


520 


634 


527 


1-365 


505 


529 


517 


1-599 


500 


515 


502 


1-815 


495 


487 


491 


1-764 


483 


495 


489 


1-581 


441 


529 


485 


1-621 


478 


470 


474 


1-512 


462 


483 


472 


1-852 


463 





463 


1-55 


466 


453 


459 


1-748 


457 


455 


456 


1-730 


453 


457 


455 


1-811 


438 


473 


455 


1-484 


453 


455 


454 


1-957 


443 


464 


453 


1-734 


441 


457 


449 


1-759 


4:^3 


4(U 


448 


1-726 


448 




448 


•790 


427 


467 


447 


1-557 


460 


434 


447 


1-825 


444 





444 


1-414 


444 


443 


443 


1-336 


456 


430 


443 


1-469 


440 


444 


442 


1-887 


430 


454 


442 


1.687 


439 


441 


440 


1-857 


432 


449 


440 


1-443 


427 


449 


438 


1-368 


4^6 





436 


1-64 


461 


403 


432 


1-516 


408 


455 


431 


1-251 


419 


439 


429 


1-358 


413 


446 


429 


1-339 


408 


446 


427 


1-512 


446 


408 


427 


2-2-24 


422 


430 


426 


1-450 


464 


385 


424 


1-532 


430 


408 


419 


1-231 


417 


419 


418 


1-570 


404 


4^2 


418 


1-222 


409 


424 


416 


1-882 


408 


418 


413 


1-470 


402 


404 


403 


1-762 


392 




392 


1-890 


353 386 


3G9 


1-525 


378 


337 


357 


1-366 



Gray- 
Whitish gray 
White 
White 

Whitish gray 
Dullish gray 
Dark gray 
Bluish gray 
Dark gray 
Gray 
Dull gray 
Dark gray 
Gray 

Light gray 
Light gray 
Dark gray 
Bluish gray 
uu^/ Dark gray 
770 Bright gray 
'^'^' Light gray 
.11 Bright gray 
353 Light gray 
998 Light gray 
841 Light gray 
629 Dark gray 
593 Gray 
674 Dull gray 
830 Bluish gray 
'-^" Gray 

Dull blue 
Light gray 
Gray 

Dull gray 
Light gray 
Light gray 
Dull gray 
Light'gray 
Bluish gray 
Gray 
Gray 

Whitish gray 
Gray- 
Bluish gray 
Dark gray 
1 . J. Brieht gray 
600 Gray 
709 Bluish gray 
742 Bluish gray 
='^'' Gray 

Light gray 



The irons with asterisks are taken from Experiments on Hot and 
Cold Blast Iron. 



THE STEAM ENaiNE. 



175 



Rule. — To find from the above Table the breaking weight in 

rectangular bars, generally. Calling h and d the breadth and 

depth in inches, and I the distance between the supports, in feet, 

4-5 xh cF jS 
and putting 4*5 for 4 ft. 6 in., we have j = breaking 

weight in lbs., — the value of S being taken from the above Table. 
For example : — What weight would be necessary to break a bar 
of Low Moor Iron, 2 inches broad, 3 inches deep, and 6 feet be- 
tween the supports ? According to the rule given above, we have 
6 = 2 inches, c? = 3 inches, I = 6 feet, S = 472 from the Table. 

^^ 4-5 xbd'^ 4-5 X 2 X 32 X 472 ^^^^ ^^ r. ^ • , 
Then — —7 = ^ = 6372 lbs., the break- 



ing weight. 
Table of the 



ive Poiver of Bodies whose 
equal one Square Inch. 



Sectional Areas 



Metals. 

Swedish bar iron 

Kussian do 

English do 

Cast steel ." 

Blistered do 

Shear do 

Wrought copper 

Hard gun-metal 

Cast copper 

Yellow brass, cast 

Cast iron 

Tin, cast 

Bismuth, cast 

Lead, cast 

Elastic power or direct tension of wroughtiron, 
medium quality 



Cohesive Power 
ia lbs. 



60,000 

59,470 

56,000 

134,256 

133,152 

127,632 

33,892 

36,368 

19,072 

17,968 

17,628 

4,736 

3,250 

1,824 

22,400 



Note. — A bar of iron is extended -000096, or nearly one ten- 
thousandth part of its length, for every ton of direct strain per 
square inch of sectional area. 



CENTRE OP GRAVITY. 

The centre of gravity of a body is that point within it which 
continually endeavours to gain the lowest possible situation ; or it is 
that point on which the body, being freely suspended, will remain 
at rest in all positions. The centre of gravity of a body does not 
always exist within the matter of which the body is composed, 
there being bodies of such forms as to preclude the possibility of 
this being the case, but it must either be surrounded by the con- 
stituent matter, or so placed that the particles shall be symmetri- 
cally situated, with respect to a vertical line in which the position 
of the centre occurs. Thus, the centre of gravity of a ring is not 
in the substance of the ring itself, but, if the ring be uniform, it will 
be in the axis of its circumscribing cylinder ; and if the ring varies 



176 THE PRACTICAL MODEL CALCULATOR. 

in form or density, it will be situated nearest to those parts where 
the weight or density is greatest. Varying the position of a body 
will not cause any change in the situation of the centre of gravity ; 
for any change of position the body undergoes will only have the 
effect of altering the directions of the sustaining forces, which will 
still preserve their parallelism. When a body is suspended by any 
other point than its centre of gravity, it will not rest unless that 
centre be in the same vertical line with the point of suspension ; 
for, in every other position, the force which is intended to insure 
the equilibrium will not directly oppose the resultant of gravity 
upon the particles of the body, and of course the equilibrium will 
not obtain ; the directions of the forces of gravity upon the con- 
stituent particles are all parallel to one another and perpendicular 
to the horizon. If a heavy body be sustained by two or more 
forces, their lines of direction must meet either at the centre of 
gravity, or in the vertical line in which it occurs. 

A body cannot descend or fall downwards, unless it be in such 
a position that by its motion the centre of gravity descends. If a 
body stands on a plane, and a line be drawn perpendicular to the 
horizon, and if this perpendicular line fall within the base of the 
body, it will be supported without falling ; but if the perpendicular 
falls without the base of the body, it will overset. For when the 
perpendicular falls within the base, the body cannot be moved at all 
without raising the centre of gravity ; but when the perpendicular 
falls without the base towards any side, if the body be moved 
towards that side, the centre of gravity will descend, and conse- 
quently the body will overset in that direction. If a perpendicular 
to the horizon from the centre of gravity fall upon the extremity 
of the base, the body may continue to stand, but the least force 
that can be applied will cause it to overset in that direction ; and 
the nearer the perpendicular is to any side the easier the body will 
be made to fall on that side, but the nearer the perpendicular is to 
the middle of the base the firmer the body will stand. If the 
centre of gravity of a body be supported, the whole body is sup- 
ported, and the place of the centre of gravity must be considered 
as the place of the body, and it is always in a line which is perpen- 
dicular to the horizon. 

In any two bodies, the common centre of gravity divides the 
line that joins their individual centres into two parts that are to 
one another reciprocally as the magnitudes of the bodies. The 
products of the bodies multiplied by their respective distances from 
the common centre of gravity are equal. If a weight be laid 
upon any point of an inflexible lever which is supported at the 
ends, the pressure on each point of the support will be inversely 
as the respective distances from the point where the weight is 
applied. In a system of three bodies, if a line be drawn from the 
centre of gravity of any one of them to the common centre of the 
other two, then the common centre of all the three bodies divides 
the line into two parts that are to each other reciprocally as the 



THE STEAM ENGINE. 177 

magnitude of the body from which the line is drawn to the sum of 
the magnitudes of the other two ; and, consequently, the single 
body multiplied by its distance from the common centre of gravity 
is equal to the sum of the other bodies multiplied by the distance 
of their common centre from the common centre of the system. 

If there be taken any point in the straight line or lever joining 
the centres of gravity of two bodies, the sum of the two products 
of each body multiplied by its distance from that point is equal to 
the product of the sum of the bodies multiplied by the distance of 
their common centre of gravity from the same point. The two 
bodies have, therefore, the same tendency to turn the lever about 
the assumed point, as if they were both placed in their common 
centre of gravity. Or, if the line with the bodies moves about the 
assumed point, the sum of the momenta is equal to the momentum 
of the sum of the bodies placed at their common centre of gravity. 
The same property holds with respect to any number of bodies 
whatever, and also when the bodies are not placed in the line, but 
in perpendiculars to it passing through the bodies. If any plane 
pass through the assumed point, perpendicular to the line in which 
it subsists, then the distance of the common centre of gravity of 
all the bodies from that plain is equal to the sum of all the 
momenta divided by the sum of all the bodies. We may here 
specify the positions of the centre of gravity in several figures of 
very frequent occurrence. 

In a straight line, or in a straight bar or rod of uniform figure 
and density, the position of the centre of gravity is at the middle 
of its length. In the plane of a triangle the centre of gravity is 
situated in the straight line drawn from any one of the angles to 
the middle of the opposite side, and at two-thirds of this line dis- 
tant from the angle where it originates, or one-third distant from 
the base. In the surface of a trapezium the centre of gravity is in 
the intersections of the straight lines that join the centres of the 
opposite triangles made by the two diagonals. The centre of 
gravity of the surface of a parallelogram is at the intersection of 
the diagonals, or at the intersection of the two lines which bisect 
the figure from its opposite sides. In any regular polygon the 
centre of gravity is at the same point as the centre of magnitude. 
In a circular arc the position of the centre of gravity is distant 
from the centre of the circle by the measure of a fourth propor- 
tional to the arc, radius, and chord. In a semicircular arc the 
position of the centre of gravity is distant from the centre by the 
measure of a third proportional to the arc of the quadrant and the 
radius. In the sector of a circle the position of the centre of 
gravity is distant from the centre of the circle by a fourth propor- 
tional to three times the arc of the sector, the chord of the arc, 
and the diameter of the circle. In a circular segment, the position 
of the centre of gravity is distant from the centre of the circle by 
a space which is equal to the cube or third power of the chord 
divided by twelve times the area of the segment. In a semicircle 

12 



178 THE PRACTICAL MODEL CALCULATOR. 

the position of the centre of gravity is distant from the centre of 
the circle by a space which is equal to four times the radius divided 
by the constant number 3-1416 X 3 = 9-4248. In a parabola the 
position of the centre of gravity is distant from the vertex by 
three-fifths of the axis. In a semi-parabola the position of the 
centre of gravity is at the intersection of the co-ordinates, one of 
which is parallel to the base, and distant from it by two-fifths of 
the axis, and the other parallel to the axis, but distant from it by 
three-eighths of the semi-base. 

The centres of gravity of the surface of a cylinder, a cone, and 
conic frustum, are respectively at the same distances from the origin 
as are the centres of gravity of the parallelogram, the triangle, and 
the trapezoid, which are sections passing along the axes of the re- 
spective solids. The centre of gravity of the surface of a spheric seg- 
ment is at the middle of the versed sine or height. The centre of 
gravity of the convex surface of a spherical zone is at the middle of 
that portion of the axis of the sphere intercepted by its two bases. 
In prisms and cylinders the position of the centre of gravity is at the 
middle of the straight line that joins the centres of gravity of their 
opposite ends. In pyramids and cones the centre of gravity is in 
the straight line that joins the vertex with the centre of gravity 
of the base, and at three-fourths of its length from the vertex, and 
one-fourth from the base. In a semisphere, or semispheroid, the 
position of the centre of gravity is distant from the centre by three- 
eighths of the radius. In a parabolic conoid the position of the 
centre of gravity is distant from the base by one-third of the axis, 
or two-thirds of the axis distant from the vertex. There are 
several other bodies and figures of which the position of the centre 
of gravity is known ; but as the position in those cases cannot be 
defined without algebra, we omit them. 

CENTRIPETAL AND CENTRIFUGAL FORCES. 

Central forces are of two kinds, centripetal and centrifugal. 
Centripetal force is that force by which a body is attracted or 
impelled towards a certain fixed point as a centre, and that point 
towards which the body is urged is called the centre of attraction 
or the centre of force^ Centrifugal force is that force by which a 
body endeavours to recede from the centre of attraction, and from 
which it would actually fly off in the direction of a tangent if it 
were not prevented by the action of the centripetal force. These 
two forces are therefore antagonistic ; the action of the one being 
directly opposed to that of the other. It is on the joint action of 
these two forces that all curvilinear motion depends. Circular motion 
is that affection of curvilinear motion where the body is constrained 
to move in the circumference of a circle : if it continues to move so 
as to describe the entire circle, it is denominated rotatory motion, and 
the body is said to revolve in a circular orbit, the centre of which is 
called the centre of motion. In all circular motions the deflection 
or deviation from the rectilinear course is constantly the same at 



THE STEAM ENGINE. 179 

every point of tlie orbit, in which case the centripetal and centri- 
fugal forces are equal to one another. In circular orbits the cen- 
tripetal forces, by which equal bodies placed at equal distances 
from the centres of force are attracted or drawn towards those 
centres, are proportional to the quantities of matter in the central 
bodies. This is manifest, for since all attraction takes place 
towards some particular body, every particle in the attracting body 
must produce its individual effect ; consequently, a body containing 
twice the quantity of matter will exert twice the attractive energy, 
and a body containing thrice the quantity of matter will operate 
with thrice the attractive force, and so on according to the quantity 
of matter in the attracting body. 

Any body, whether large or small, when placed at the same dis- 
tance from the centre of force, is attracted or drawn through equal 
spaces in the same time by the action of the central body. This 
is obvious from the consideration that although a body two or three 
times greater is urged with two or three times greater an attractive 
force, yet there is two or three times the quantity of matter to be 
moved ; and, as we have shown elsewhere, the velocity generated 
in a given time is directly proportional to the force by which it is 
generated, and inversely as the quantity of matter in the moving 
or attracted body. But the force which in the present instance is 
the weight of the body is proportional to the quantity of matter 
which it contains ; consequently, the velocity generated is directly 
and inversely proportional to the quantity of matter in the 
attracted body, and is, therefore, a given or a constant quantity. 
Hence, the centripetal force, or force towards the centre of the 
circular orbit, is not measured by the magnitude of the revolving 
body, but only by the space which it describes or passes over in a 
given time. When a body revolves in a circular orbit, and is 
retained in it by means of a centripetal force directed to the 
centre, the actual velocity of the revolving body at every point of 
its revolution is equal to that which it would acquire by falling 
perpendicularly with the same uniform force through one-fourth of 
the diameter, or one-half the radius of its orbit ; and this velocity 
is the same as would be acquired by a second body in falling 
through half the radius, whilst the first body, in revolving in its 
orbit, describes a portion of the circumference which is equal in 
length to half the diameter of the circle. Consequently, if a body 
revolves uniformly in the circumference of a circle by means of a 
given centripetal force, the portion of the circumference which it 
describes in any time is a mean proportional between the diameter 
of the circle and the space which the body would descend perpen- 
dicularly in the same time, and with the same given force continued 
uniformly. 

The feriodie time, in the doctrine of central forces, is the time 
occupied by a body in performing a complete revolution round the 
centre, when that body is constrained to move in the circumference 
by means of a centripetal force directed to that point ; and when 



180 THE PRACTICAL MODEL CALCULATOR. 

the body revolves in a circular orbit, the periodic time, or the 
time of performing a complete revolution, is expressed by the term 

?t f v/ -5 and the velocity or space ^passed over in the time t will be 

y/ d§', in which expressions d denotes the diameter of the circular 
orbit described by the revolving body, s the space descended in any 
time by a body falling perpendicularly downwards with the same 
Uniform force, t the time of descending through the space, s and n 
the circumference of a circle whose diameter is unity. If several 
bodies revolving in circles round the same or different centres be 
retained in their orbits by the action of centripetal forces directed 
to those points, the periodic times will be directly as the square 
roots of the radii or distances of the revolving bodies, and inversely 
as the square roots of the centripetal forces, or, what is the same 
thing, the squares of the periodic times are directly as the radii, 
and inversely as the centripetal forces. 

CENTRE OF GYRATION. 

The centre of gyration is that point in which, if all the consti- 
tuent particles, or all the matter contained in a revolving body, or 
system of bodies, were concentrated, the same angular velocity 
would be generated in the same time by a given force aeting at any 
place as would be generated by the same force acting similarly on 
the body or system itself according to its formation. 

The angular motion of a body, or system of bodies, is the motion 
of a line connecting any point with the centre or axis of motion, 
and is the same in all parts of the same revolving system. 

In different unconnected bodies, each revolving about a centre, 
the angular velocity is directly proportional to the absolute velo- 
city, and inversely as the distance from the centre of motion ; so 
that, if the absolute velocities of the revolving bodies be propor- 
tional to their radii or distances, the angular velocities will be 
equal. If the axis of motion passes through the centre of gravity, 
then is this centre called the principal centre of gyration. 

The distance of the centre of gyration from the point of suspen- 
sion, or the axis of motion in any body or system of bodies, is a 
geometrical mean between the centres of gravity and oscillation 
from the same point or axis ; consequently, having found the dis- 
tances of these centres in any proposed case, the square root of 
their product will give the distance of the centre of gyration. If 
any part of a system be conceived to be collected in the centre of 
gyration of that particular part, the centre of gyration of the 
whole system will continue the same as before ; for the same force that 
moved this part of the system before along with the rest will move 
it now without any change ; and consequently, if each part of the 
system be collected into its own particular centre, the common 
centre of the whole system will continue the same. If a circle be 
described about the centre of gravity of any system, and the axis 
of rotation be made to pass through any point of the circumference, 



THE STEAM ENGINE. 181 

the distance of the centre of gyration from that point will always 
be the same. 

If the periphery of a circle revolve about an axis passing through 
the centre, and at right angles to its plane, it is the same thing as 
if all the matter were collected into any one point in the peri- 
phery. And moreover, the plane of a circle or a disk containing 
twice the quantity of matter as the said periphery, and having the 
same diameter, will in an equal time acquire the same angular 
velocity. If the matter of a revolving body were actually to be 
placed in the centre of gyration, it ought either to be arranged in 
the circumference, or in two points of the circumference diametri- 
cally opposite to each other, and equally distant from the centre 
of motion, for by this means the centre of motion will coincide 
with the centre of gravity, and the body will revolve without any 
lateral force on any side. These are the chief properties con- 
nected with the centre of gyration, and the following are a few of 
the cases in which its position has been ascertained. 

In a right line, or a cylinder of very small diameter revolving 
about one of its extremities, the distance of the centre of gyration 
from the centre of motion is equal to the length of the revolving 
line or cylinder multiplied by the square root of J. In the plane 
of a circle, or a cylinder revolving about the axis, it is equal to the 
radius multiplied by the square root of ^. In the circumference 
of a circle revolving about the diameter it is equal to the radius 
multiplied by the square root of J. In the plane of a circle 
revolving about the diameter it is equal to one-half the radius. In 
a thin circular ring revolving about one of its diameters as an axis 
it is equal to the radius multiplied by the square root of ^. In a 
solid globe revolving about the diameter it is equal to the radius 
multiplied by the square root of |. In the surface of a sphere 
revolving about the diameter it is equal to the radius multiplied by 
the square root of f . In a right cone revolving about the axis it 
is equal to the radius of the base multiplied by the square root of 3^. 
In all these cases the distance is estimated from the centre of 
the axis of motion. We shall have occasion to illustrate these prin- 
ciples when we come to treat of fly-wheels in the construction of 
the different parts of steam engines. 

When bodies revolving in the circumferences of different circles 
are retained in their orbits by centripetal forces directed to the 
centres, the periodic times of revolution are directly proportional 
to the distances or radii of the circles, and inversely as the veloci- 
ties of motion ; and the periodic times, under like circumstances, 
are directly as the velocities of motion, and inversely as the cen- 
tripetal forces. If the times of revolution are equal, the velocities 
and centripetal forces are directly as the distances or radii of the 
circles. If the centripetal forces are equal, the squares of the 
times of revolution and the squares of the velocities are as the dis- 
tances or radii of the circles. If the times of revolution are as 
Q 



182 THE PKACTICAL MODEL CALCULATOR. 

the radii of the circles, the velocities will be equal, and the cen- 
tripetal forces reciprocally as the radii. 

If several bodies revolve in circular orbits round the same or 
difierent centres, the velocities are directly as the distances or 
radii, and inversely as the times of revolution. The velocities are 
directly as the centripetal forces and the times of revolution. The 
squares of the velocities are proportional to the centripetal forces, 
and the distances or radii of the circles. When the velocities are 
equal, the times of revolution are proportional to the radii of the 
circles in which the bodies revolve, and the radii of the circles are 
inversely as the centripetal forces. If the velocities be propor- 
tional to the distances or radii of the circles, the centripetal forces 
will be in the same ratio, and the times of revolution will be equal. 

If several bodies revolve in circular orbits about the same or 
different centres, the centripetal forces are proportional to the dis- 
tances or radii of the circles directly, and inversely as the squares 
of the times of revolution. The centripetal forces are directly 
proportional to the velocities, and inversely as the times of revolu- 
tion. The centripetal forces are directly as the squares of the 
velocities, and inversely as the distances or radii of the circles. 
When the centripetal forces are equal, the velocities are propor- 
tional to the times of revolution, and the distances as the squares 
of the times or as the squares of the velocities. When the central 
forces are proportional to the distances or radii of the circles, the 
times of revolution are equal. If several bodies revolve in circular 
orbits about the same or different centres, the radii of the circles 
are directly proportional to the centripetal forces, and the squares 
of the periodic times. The distances or radii of the circles are 
directly as the velocities and periodic times. The distances or 
radii of the circles are directly as the squares of the velocities, and 
reciprocally as the centripetal forces. If the distances are equal, 
the centripetal forces are directly as the squares of the velocities, 
and reciprocally as the squares of the times of revolution ; the 
velocities also are reciprocally as the times of revolution. The 
converse of these principles and properties are equally true ; and 
all that has been here stated in regard to centripetal forces is 
similarly true of centrifugal forces, they being equal and contrary 
to each other. 

The quantities of matter in all attracting bodies, having other 
bodies revolving about them in circular orbits, are proportional to 
the cubes of the distances directly, and to the squares of the times 
of revolution reciprocally. The attractive force of a body is 
directly proportional to the quantity of matter, and inversely as 
the square of the distance. If the centripetal force of a body 
revolving in a circular orbit be proportional to the distance from 
the centre, a body let fall from the upper extremity of the vertical 
diameter will reach the centre in the same time that the revolving 
body describes one-fourth part of the orbit. The velocity of the 
descending body at any point of the diameter is proportional to 



THE STEAM ENGINE. 183 

the ordinate of the circle at that point ; and the time of falling 
through any portion of the diameter is proportional to the arc of 
the circumference whose versed sine is the space fallen through. 
All the times of falling from any altitudes whatever to the centre 
of the orbit will be equal ; for these times are equal to one-fourth 
of the periodic times, and these times, under the specified condi- 
tions, are equal. The velocity of the descending body at the centre 
of the circular orbit is equal to the velocity of the revolving body. 
These are the chief principles that we need consider regarding 
the motion of bodies in circular orbits ; and from them we are led 
to the consideration of bodies suspended on a centre, and made to 
revolve in a circle beneath the suspending point, so that Avhen the 
body describes the circumference of a circle, the string or wire by 
which it is suspended describes the surface of a cone. A body thus 
revolving is called a conical pendulum^ and this species of pendu- 
lum, or, as it is usually termed, the governor, is of great importance 
in mechanical arrangements, being employed to regulate the move- 
ments of steam engines, water-wheels, and other mechanism. As 
we shall have occasion to show the construction and use of this in- 
strument when treating of the parts and proportions of engines, we 
need not do more at present than state the principles on which its 
action depends. We must, however, previously say a few Avords 
on the properties of the simple pendulum, or that which, being sus- 
pended from a centre, is made to vibrate from side to side in the 
same vertical plane. 

PENDULUMS. 

If a pendulum vibrates in a small circular arc, the time of per- 
forming one vibration is to the time occupied by a heavy body in 
falling perpendicularly through half the length of the pendulum as 
the circumference of a circle is to its diameter. All vibrations of 
the same pendulum made in very small circular arcs, are made in 
very nearly the same time. The space described by a falling body 
in the time of one vibration is to half the length of the pendulum 
as the square of the circumference of a circle is to the square of 
the diameter. The lengths of two pendulums which by vibrating 
describe similar circular arcs are to each other as the squares of 
the times of vibration. The times of pendulums vibrating in small 
circular arcs are as the square roots of the lengths of the pendulums. 
The velocity of a pendulum at the lowest point of its path is pro- 
portional to the chord of the arc through which it descends to ac- 
quire that velocity. Pendulums of the same length vibrate in the 
same time, whatever the weights may be. From which we infer, 
that all bodies near the earth's surface, whether they be heavy or 
light, will fall through equal spaces in equal times, the resistance 
of the air not being considered. 

The lengths of pendulums vibrating in the same time in different 
positions of the earth's surface are as the forces of gravity in those 
positions. The times wherein pendulums of the same length will 
vibrate by different forces of gravity are inversely as the square 



184 THE PRACTICAL MODEL CALCULATOR. 

roots of the forces. The lengths of pendulums vibrating in dif- 
ferent places are as the forces of gravity at those places and the 
squares of the times of vibration. The times in Tvhich pendulums 
of any length perform their vibrations are directly as the square 
roots of their lengths, and inversely as the square roots of the gravi- 
tating forces. The forces of gravity at different places on the earth's 
surface are directly as the lengths of the pendulums, and inversely 
as the squares of the times of vibration. These are the chief proper- 
ties of a simple pendulum vibrating in a vertical plane, and the prin- 
cipal problems that arise in connection with it are the following, viz. : 

To find the length of a pendulum that shall make any number 
of vih^ations in a given time ; and secondly, having given the length 
of a pendulum, to find the number of vibrations it tvill make in any 
time given. — These are problems of very easy solution, and the 
rules for resolving them are simply as follow : — For the first, the 
rule is, multiply the square of the number of seconds in the given 
time by the constant number 39*1015, and divide the product by 
the square of the number of vibrations, for the length of the 
pendulum in inches. For the second, it is, multiply the square of 
the number of seconds in the given time by the constant number 
39-1393, divide the product by the given length of the pendulum 
in inches, and extract the square root of the quotient for the num- 
ber of vibrations sought. The number 39*1015 is the length of a 
pendulum in inches, that vibrates seconds, or sixty times in a minute, 
in the latitude of Philadelphia. 

Suppose a pendulum is found to make 35 vibrations in a minute ; 
what is the distance from the centre of suspension to the centre of 
oscillation ? 

Here, by the rule, the number of seconds in the given time is 60 ; 
hence we get 60 X 60 X 39*1015 = 140765*4, which, being di- 
vided by 35 X 35 = 1225, gives 140765*4 -^ 1225 = 114*9105 
inches for the length required. 

The length of a pendulum between the centre of suspension and 
the centre of oscillation is 64 inches ; what number of vibrations 
will it make in 60 seconds ? 

By the rule we have 60 X 60 X 39*1015 = 140765*4, which, 
being divided by 64, gives 140765*4 -- 64 = 2199*46, and the 
square root of this is 2199*46 = 46*9, number of vibrations 
sought. When the given time is a minute, or 60 seconds, as in the 
two examples proposed above, the product of the constant number 
39*1015 by the square of the time, or 140765*4, is itself a constant 
quantity, which, being kept in mind, will in some measure facilitate the 
process of calculation in all similar cases. We now return to the 
consideration of the conical pendulum, or that in which the ball re- 
volves about a vertical axis in the circumference of a circular plane 
which is parallel to the horizon. 

CONICAL PENDULUM. 

If a pendulum be suspended from the upper extremity of a ver- 
tical axis, and be made to revolve about that axis by a conical mo- 




THE STEAM ENGINE. 185 

tion, which constrains the revolving body to move in the circum- 
ference of a circle whose plane is parallel to the horizon, then the 
time in which the pendulum performs a revolution about the axis 
can easily be found. 

Let CD be the pendulum in question, suspended from C, the 
upper extremity of the vertical axis CD, 
and let the ball or body B, by revolving 
about the said axis, describe the circle BE 
AH, the plane of which is parallel to the 
horizon ; it is proposed to assign the time 
of description, or the time in which the body 
B performs a revolution about the axis CD, 
at the distance BD. 

Conceive the axis CD to denote the weight' 
of the revolving body, or its force in the di- 
rection of gravity; then, by the Compo- 
sition and Resolution of Forces, CB will denote the force or 
tension of the string or wire that retains the revolving body in 
the direction CB, and BD the force tending to the centre of the 
plane of revolution at D. But, by the general laws of motion 
and forces previously laid down, if the time be given, the space 
described will be directly proportional to the force ; but, by the 
laws of gravity, the space fallen perpendicularly from rest, in one 
second of time, is ^ = 16^ feet ; consequently we have CD : BD : : 

16^2 : ^' — , the space described towards D by the force in BD 

in one second. Consequently, by the laws of centripetal forces, the 
periodic time, or the time of the body revolving in the circle BEAH, 

|2'CD 

is expressed by the term 7t^ ., where 7t = 3-1416, the circum- 
ference of a circle whose diameter is unity ; or putting t to denote 
the time, and expressing the height CD in feet, we get t = 6-2832 

^/-- — ^— , or, by reducing the expression to its simplest form, it 

becomes t = 0-31986v/CD, where CD must be estimated in inches, 
and t in seconds. Here we have obtained an expression of great 
simplicity, and the practical fule for reducing it may be expressed 
in words as follows : 

Rule. — Multiply the square root of the height, or the distance 
between the point of suspension and the centre of the plane of revo- 
lution, in inches, by the constant fraction 0-31986, and the product 
will be the time of revolution in seconds. 

In what time will a conical pendulum revolve about its vertical 
axis, supposing the distance between the point of suspension and 
the centre of the plane of revolution to be 39*1393 inches, which is 
the length of a simple pendulum that vibrates seconds in latitude 
51° 30' ? 

The square root of 39*1393 is 6*2561 ; consequently, by the rule, 
q2 



186 



THE PRACTICAL MODEL CALCULATOR. 



we have, 6-2561 X 0-31986 = 2-0011 seconds for the time of revo- 
lution sought. It consequently revolves 30 times in a minute, as it 
ought to do by the theory of the simple pendulum. 

By reversing the process, the height of the cone, or the distance 
between the point of suspension and the centre of the plane of revo- 
lution, corresponding to any given time, can easily be ascertained ; 
for we have only to divide the number of seconds in the given time 
by the constant decimal 0-31986, and the square of the quotient 
will be the required height in inches. ' Thus, suppose it were re- 
quired to find the height of a conical pendulum that would revolve 
30 times in a minute. Here the time of revolution is 2 seconds for 
60 -7- 30 = 2; therefore, by division, it is 2 h- 0-31986 = 6-2527, 
which, being squared, gives 6-2527 = 39*0961 inches, or the length 
of a simple pendulum that vibrates seconds very nearly. In all 
conical pendulums the times of revolution, or the periodic times, are 
proportional to the square roots of the heights of the cones. This 
is manifest, for in the foregoing equation of the periodic time the 
numbers 6-2832 and 386, or 12 x 32i, are constant quantities, con- 
sequently t varies as n/CD. 

If the heights of the cones, or the distances between the points 
of suspension and the centres of the planes of revolution, be the 
same, the periodic times, or the times of revolution, will be the 
same, whatever may be the radii of the circles described by the re- 




volving bodies. This will be clearly understood by contemplating 
the subjoined diagram, where all the pendulums C«, C5, C(?, Qd, and 
Qe, having the common axis CD, will revolve in the same time ; and 



THE STEAM ENGINE. 187 

if they are all in the same vertical plane when first put in motion, 
they will continue to revolve in that plane, whatever be the velocity, 
so long as the common axis or height of the cone remains the same. 
This will become manifest, if we conceive an inflexible bar or rod 
of iro^ to pass through the centres of all the balls as well as the 
common axis, for then the bar and the several balls must all revolve 
in the same time ; but if any one of them should be allowed to rise 
higher, its velocity would be increased ; and if it descends, the ve- 
locity will be decreased. 

Half the periodic time of a conical pendulum is equal to the 
time of vibration of a simple pendulum, the length of which is 
equal to the axis or height of the cone ; that is, the simple pendu- 
lum makes two oscillations or vibrations from side to side, or it 
arrives at the same point from which it departed, in the same time 
that the conical pendulum revolves about its axis. The space 
descended by a falling body in the time of one revolution of the 
conical pendulum is equal to 3-1416^ multiplied by twice the height 
or axis of the cone. The periodic time, or the time of one revo- 
lution is equal to the product of 3*1416 y/ 2 multiplied by the time 
of falling through the height of the cone. The weight of a conical 
pendulum, when revolving in the circumference of a circle, bears 
the same proportion to the centrifugal force, or its tendency to fly 
off in a straight line, as the axis or height of the cone bears to the 
radius of the plane of revolution ; consequently, when the height 
of the cone is equal to the radius of its base, the centripetal or 
centrifugal force is equal to the power of gravity. 

These are the principles on which the action of the conical pen- 
dulum depends ; but as we shall hereafter have occasion to con- 
sider it more at large, we need not say more respecting it in this 
place. Before dismissing the subject, however, it may be proper to 
put the reader in possession of the rules for calculating the posi- 
tion of the centre of oscillation in vibrating bodies, in a few cases 
where it has been determined, these being the cases that are of the 
most frequent occurrence in practice. 

The ce7itre of oscillation in a vibrating body is that point in the 
line of suspension, in which, if all the matter of the system were 
collected, any force applied there would generate the same angular 
motion in a given time as the same force applied at the centre of 
gravity. The centres of oscillation for several figures of very fre- 
quent use, suspended from their vertices and vibrating flatwise, are 
as follow : — 

In a right line, or parallelogram, or a cylinder of very small 
diameter, the centre of oscillation is at two-thirds of the length 
from the point of suspension. In an isosceles triangle the centre 
of oscillation is at three-fourths of the altitude. In a circle it 
is five-fourths of the radius. In the common parabola it is 
five-sevenths of its altitude. In a parabola of any order it is 
/2 rt -f 1\ 
\^o — jlTT/ ^ altitude, where n denotes the order of the figure. 



188 THE PRACTICAL MODEL CALCULATOR. 

In bodies vibrating laterally, or in their own plane, tbe centres 
of oscillation are situated as follows ; namely, in a circle the centre 
of oscillation is at three-fourths of the diameter ; in a rectangle, 
suspended at one of its angles, it is at two-thirds of the diagonal ; 
in a parabola, suspended by the vertex, it is five-sevenths of the 
axis, increased by one-third of the parameter ; in a parabola, sus- 
pended by the middle of its base, it is four-sevenths of the axis, 
increased by half the parameter; in the sector of a circle it is 
three times the arc of the sector multiplied by the radius, and 
divided by four times the chord ; in a right cone it is four-fifths of 
the axis or height, increased by the quotient that arises when the 
square of the radius of the base is divided by five times the height ; 
in a globe or sphere it is the radius of the sphere, plus the length of 
the thread by which it is suspended, plus the quotient that arises 
when twice the square of the radius is divided by five times the sum 
of the radius and the length of the suspending thread. In all these 
cases the distance is estimated from the point of suspension, and since 
the centres of oscillation and percussion are in one and the same 
point, whatever has been said of the one is equally true of the other. 

THE TEMPERATURE AND ELASTIC EORCE OP STEAM, 

In estimating the mechanical action of steam, the intensity of its 
elastic force must be referred to some known standard measure, 
such as the pressure which it exerts against a square inch of the 
surface that contains it, usually reckoned by so many pounds 
avoirdupois upon the square inch. The intensity of the elastic 
force is also estimated by the inches in height of a vertical column 
of mercury, whose weight is equal to the pressure exerted by the 
steam on a surface equal to the base of the mercurial column. It 
may also be estimated by the height of a vertical column of water 
measured in feet ; or generally, the elastic force of any fluid may 
be compared with that of atmospheric air when in its usual state of 
temperature and density ; this is equal to a column of mercury 30 
inches or 2| feet in height. 

When the temperature of steam is increased, respect being had 
to its density, the elastic force, or the effort to separate the parts 
of the containing vessel and occupy a larger space, is also increased ; 
and when the temperature is diminished, a corresponding and pro- 
portionate diminution takes place in the intensity of the emanci- 
pating effort or elastic power. It consequently follows that there 
must be some law or principle connecting the temperature of steam 
with its elastic force ; and an intimate acquaintance with this law, 
in so far as it is known, must be of the greatest importance in all 
our researches respecting the theory and the mechanical operations 
of the steam engine. 

To find a theorem, hy means of ivliich it may he ascertained when 
a general law exists, and to determine what that laiv is, in cases 
where it is knoivn to oUain. — Suppose, for example, that it is 
required to assign the nature of the law that subsists between the 



THE STEAM ENGINE. 18Q 

temperature of steam and its elastic force, on tlie supposition that 
the elasticity is proportional to some power of the temperature, 
and unaffected by any other constant or co-efficient, except the 
exponent by which the law is indicated. Let E and e be any two 
values of the elasticity, and T, ^, the corresponding temperatures 
deducted from observation. It is proposed to ascertain the powers 
of T and t^ to which E and e are respectively proportional. Let n 
denote the index or exponent of the required power ; then by the 
conditions of the problem admitting that a law exists, we get, 

T" : r : : E : e ; but by the principles of proportion, it is — = — ; 

t e 

and if this be expressed logarithmically, it is w X log. — = log. — , 

T E 

and by reducing the equation in respect of w, it finally becomes 
_ log. e — log. E 
'' ~ log. t - log. T* 

The theorem that we have here obtained is in its form suffi- 
ciently simple for practical application ; it is of frequent occur- 
rence in physical science, but especially so in inquiries respecting 
the motion of bodies moving in air and other resisting media ; and 
it is even applicable to the determination of the planetary motions 
themselves. The process indicated by it in the case that we have 
chosen, is simply. To divide the difference of the logarithms of the 
elasticities hy the difference of the logarithms of the corresponding 
temperatures^ and the quotient will express that power of the tempe- 
rature to which the elasticity is proportional. 

Take as an example the following data : — In two experiments it 
was found that when the temperature of steam was 250*3 and 
343-6 degrees of Fahrenheit's scale, the corresponding elastic 
forces were 59*6 and 238-4 inches of the mercurial column respec- 
tively. From these data it is required to determine the law which 
connects the temperature with the elastic force on the supposition 
that a law does actually exist under the specified conditions. The 
process by the rule is as follows : 

Greater temperature, 343-6 log. 2-5352941 

Lesser temperature, 250-3 log. 2-3984608 

Remainder = 0-1368333 

G-reater elastic force, 238*4 log. 2-3773063 

Lesser elastic force, 59-6 log. 1-7752463 

Remainder = 0-6020600 

Let the second of these remainders be divided by the first, as 
directed in the rule, and we get n = 6020600 -r- 1368333 = 4-3998, 
the exponent sought. Consequently, by taking the nearest unit, 
for the sake of simplicity, we shall have, according to this result, 
the following analogy, viz. : 

T^^•r'4:;E :e; 



190 THE PRACTICAL MODEL CALCULATOR. 

that is, the elasticities are proportional to the 4*4 power of the 
temperatures very nearly. 

Now this law is rigorously correct, as applied to the particular 
cases that furnished it ; for if the two temperatures and one elas- 
ticity be given, the other elasticity will be found as indicated by 
the above analogy ; or if the two elasticities and one temperature 
be given, the other temperature will be found by a similar process. 
It by no means follows, however, that the principle is general, nor 
could we venture to aflSrm that the exponent here obtained will 
accurately represent the result of any other experiments than 
those from which it is deduced, whether the temperature be higher 
or lower than that of boiling water ; but this we learn from it, that 
the index which represents the law of elasticity is of a very high 
order, and that the general equation, whatever its form may be, 
must involve other conditions than those which we have assumed in 
the foregoing investigation. The theorem, however, is valuable to 
practical men, not only as being applicable to numerous other 
branches of mechanical inquiry, but as leading directly to the 
methods by which some of the best rules have been obtained for 
calculating the elasticity of steam, when in contact with the liquid, 
from which it is generated. 

We now proceed to apply our formula to the determination of a 
general law, or such as will nearly represent the class of experi- 
ments on which it rests ; and for this purpose we must first assign 
the limits, and then inquire under what conditions the limitations 
take place, for by these limitations we must in a great measure be 
guided in determining the ultimate form of the equation which 
represents the law of elasticity. 

The limits of elasticity will be readily assigned from the follow- 
ing considerations, viz. : In the first place, it is obvious that steam 
cannot exist when the cohesive attraction of the particles is of 
greater intensity than the repulsive energy of the caloric or matter 
of heat interposed between them ; for in this case, the change from 
an elastic fluid to a solid may take place without passing through 
the intermediate stage of liquidity : hence we infer that there must 
be a temperature at which the elastic force is nothing, and this 
temperature, whatever may be its value, corresponds to the lower 
limit of elasticity. The higher limit will be discovered by similar 
considerations, for it must take place when the density of steam is 
the same as that of water, which therefore depends on the modulus 
of elasticity of water. The modulus of elasticity of any substance 
is the measure of its elastic force ; that of water at 60° of tempe- 
rature is 22,100 atmospheres. Thus, for instance, suppose a given 
quantity of water to be confined in a close vessel which it exactly 
fills, and let it be exposed to a high degree of temperature, then 
it is obvious that in this state no steam would be produced, and the 
force which is exerted to separate the parts of the vessel is simply 
the expansive force of compressed water ; we therefore have the 
following proportion. As the expanded volume of water is to the 



THE STEAM ENGINE. 191 

quantity of expansion, so is the modulus of elasticity of water to 
the elastic force of steam of the same density as water. 

Having therefore assigned the limits beyond which the elastic 
force of steam cannot reach, we shall now proceed to apply the 
principle of our formula to the determination of the general law 
which connects the temperature with the elastic force ; and for this 
purpose, in addition to the notation which we have already laid 
down, let c denote some constant quantity that affects the elasticity, 
and d the temperature at which the elasticity vanishes ; then since 
this temperature must be applied subtractively, we have from the 
foregoing principle, c E = (T — 5)", and ce= {t — §)". From 
either of these equations, therefore, the constant quantity c can 
be determined in terms of the rest when they are known ; thus we 

have e = ^ — — — ^, and c = ^ '—, and by comparing these 

two independent values of c, the value of n becomes known ; for 

(T _ a)" _{t- a). 



^ and consequently 
loo-, e — I02;. E 



(A). 



log. (« - 8) - log. (T - «)_. 
In this equation the value of the symbol 5 is unknown ; in order 
therefore to determine it, we must have another independent 
expression for the value of n ; and in order to this, let the elasti- 
cities E and e become E' and e' respectively; while the corre- 
sponding temperatures T and t assume the values T' and t' ; then 

by a similar process to the above, we get ^ — ^ = ^ — - — '-, and 
n = log, e' - l og. E^ .^. 

log. (f -8)- log. (T' - 8) ; • • ^ > 

Let the equations (A) and (B) be compared with each other, and 
we shall then have an expression involving only the unknown 
quantity 5, for it must be understood that the several temperatures 
with their corresponding elasticities are to be deduced from experi- 
ment ; and in consequence, the law that we derive from them must 
be strictly empirical ; thus we have 

log, e — log. E. ^ log, e' — log. E ,^. 

log. (i - 6) - log. (T - 6) ~ log. (f'- 8) - log. (T' - 5) • ; ^ ^' 
We have no direct method of reducing expressions of this sort, 
and the usual process is therefore by approximation, or by the rule 
of trial and error, and it is in this way that the value of the quan- 
tity 8 must be found ; and for the purpose of performing the reduc- 
tion, we shall select experiments performed with great care, and 
may consequently be considered as representing the law of elas- 
ticity with very great nicety. 

T = 212-0 Fahrenheit E = 29-8 inches of mercury. 
t = 250-3 e = 59-6 

T'= 293-4 E'= 119-2 

i'= 343-6 e'= 238-4 



192 THE PRACTICAL MODEL CALCULATOR. 

Therefore, by substituting tbese numbers in equation (C), and 
making a few trials, we find that 8 = — 50°, and substituting this 
in either of the equations (A) or (B), we get n = 5*08 ; and 
finally, by substituting these values of S and n in either of the 
expressions for the constant quantity <?, we get c = 64674730000, 
the 5-08 root of which is 134-27 very nearly; hence we have 



jt±^y'\ . . .(D). 

1134-27/ ^ ^ 



Where the symbol F denotes generally the elastic force of the 
steam in inches of mercury, and t the corresponding temperature 
in degrees of Fahrenheit's thermometer, the logarithm of the J 
denominator of the fraction is 2*1279717, which may be used as a i 
constant in calculating the elastic force corresponding to any given 
temperature. We have thus discovered a rule of a very simple , 
form ; it errs in defect ; but this might have been remedied by i 
assuming two points near one extremity of the range of experi- ' 
ment, and two points near the other extremity ; and by substi- 
tuting the observed numbers in equation (C), difi'erent constants 
and a more correct exponent would accordingly have been obtained. 
Mr. Southern has, by pursuing a method somewhat analogous to 
that which is here described, found his experiments to be very 
nearly represented by 



I 135-767 / 



But even here the formula errs in defect, for he has found it 
necessary to correct it by adding the arbitrary decimal 0*1; and 
thus modified, it becomes 

I 135-767 / ^ ^ 

Our own formula may also be corrected by the application of 
some arbitrary constant of greater magnitude ; but as our motive 
for tracing the steps of investigation in the foregoing case was to 
exemplify the method of determining the law of elasticity, our end 
is answered ; for we consider it a very unsatisfactory thing merely 
to be put in possession of a formula purporting to be applicable to 
some particular purpose, without at the same time being put in 
possession of the method by which that formula was obtained, and 
the principles on which it rests. Having thus exhibited the prin- 
ciples and the method of reduction, the reader will have greater 
confidence as regards the consistency of the processes that he may 
be called upon to perform. The operation implied by equation (E) 
may be expressed in words as follows : — 

Rule. — To the given temperature in degrees of Fahrenheit's 
thermometer add 51-3 degrees and divide the sum by 135-767; to 
the 5-13 power of the quotient add the constant fraction ^, and 
the sum will be the elastic force in inches of mercury. 



THE STEAM ENGINE. 193 

The process here described is that which is performed by the 
rules of common arithmetic ; but since the index is affected by a 
fraction, it is difficult to perform in that way : we must therefore 
have recourse to logarithms as the only means of avoiding the diffi- 
culty. The rule adapted to these numbers is as follows : — 

Rule for Logarithms. — To the given temperature in degrees 
of Fahrenheit's thermometer add 51'3 degrees ; then, from the 
logarithm of the sum subtract 2*1327940 or the logarithm of 
135 '767, the denominator of the fraction ; multiply the remainder 
by the index 5*13, and to the natural number answering to the 
sum add the constant fraction Jq; the sum will be the elastic force 
in inches of mercury. 

If the temperature of steam be 250*3 degrees as indicated by 
Fahrenheit's thermometer, what is the corresponding elastic force 
in inches of mercury ? 
By the rule it is 250*3 + 51*3 = 301*6 log. 2*4794313 

constant den. = 135*767 log. 2*1327940 subtract 

remainder = 0*3466373 

31*5 inverted 

17331865 
346637 
103991 



natural number 60*013 log. 1*7782493 
If this be increased by ^"5, we get 60*113 inches of mercury for 
the elastic force of steam at 250*3 degrees of Fahrenheit. 

By simply reversing the process or transposing equation (E), the 
temperature corresponding to any given elastic force can easily be 
found ; the transformed expression is as follows, viz. : 

t = 135-767 (F - 0*1)^ - 51*3 .... (F). 

Since, in consequence of the complicated index, the process of 
calculation cannot easily be performed by common arithmetic, it is 
needless to give a rule for reducing the equation in that way ; we 
shall therefore at once give the rule for performing the process by 
logarithms. 

Rule. — From the given elastic force in inches of mercury, sub- 
tract the constant fraction 0*1 ; divide the logarithm of the remain- 
der by 5*13, and to the quotient add the logarithm 2*1327940 ; find 
the natural number answering to the sum of the logarithms, and 
from the number thus found subtract the constant 51*3, and the 
remainder will be the temperature sought. 

Supposing the elastic force of steam or the vapour of water to 
be equivalent to the weight of a vertical column of mercury, the 
height of which is 238*4 inches; what is the corresponding tem- 
perature in degrees of Fahrenheit's thermometer ? 

Here, by proceeding as directed in the rule, we have 238*4 —0*1 = 
R 13 



194 THE PRACTICAL MODEL CALCULATOR. 

238*3, and dividing the logarithm of this remainder by the con- 
stant exponent 5'13, we get 

log. 238-3 ^ 5-13 = 2-3771240 -- 5-13 = 0-4633770 
constant co-efficient = 135-767 - - log. 2-1327940 add 

natural number = 394-61 - - - log. 2-5961710 sum 

constant temperature = 51*3 subtract 

required temperature = 343-31 degrees of Fahrenheit's ther- 
mometer. 

The temperature by observation is 343*6 degrees, giving a differ- 
ence of only 0-29 of a degree in defect. For low temperature or 
low pressure steam, that is, steam not exceeding the simple pres- 
sure of the atmosphere, M. Pambour gives 

/ 1 4-51*3 \^'^^ 
i' = 0-04948 + (j55;^^) . . .(G). 

In which equation the symbol p denotes the pressure in pounds 
avoirdupois per square inch, and t the temperature in degrees of 
Fahrenheit's thermometer. When this expression is reduced in 
reference to temperature, it is 

t = 155*7256 {p - 0*04948) ^ - 51*3 .... (H), 
The formula of Tredgold is well known. The equation, in its 
original form, is 

177/i = ^ -MOO . . . . (I) : 

where / denotes the elastic force of steam in inches of mercury, 
and t the temperature in degrees of Fahrenheit's thermometer. 
The same formula, as modified and corrected by M. Millet, becomes 

179*0773/* = ^ -f 103 . . . . (K). 
Dr. Young of Dublin constructed a formula which was adapted 
to the experiments of his countryman Dr. Dalton : it assumed a 
form sufficiently simple and elegant ; it is thus expressed — 
/ = (1 + 0.0029 ^y . . . . (L): 

where the symbol / denotes the elastic force of steam expressed in 
atmospheres of 30 inches of mercury, and t the temperature in 
degrees estimated above 212 of Fahrenheit. This formula is not 
applicable in practice, especially in high temperatures, as it deviates 
very widely and rapidly from the results of observation : it is 
chiefly remarkable as being made the basis of a numerous class of 
theorems somewhat varied, but of a more correct and satisfactory 
character. The Commission of the French Academy represented 
their experiments by means of a formula constructed on the same 
principles : it is thus expressed — 

/=(1 + 0-n5Sty . . . . (M): 
where / denotes the elastic force of the steam expressed in atmo- 
spheres of 0*76 metres or 29*922 inches of mercury, and t the tern- 



THE STEAM ENGINE. 195 

perature estimated above 100 degrees of the centigrade thermo- 
meter ; but when the same formula is so transformed as to be 
expressed in the usual terms adopted in practice, it is 

p = (0-2679 + 0-0067585^)^ . . . . (N): 

where p is the pressure in pounds per square inch, and t the tem- 
perature in degrees of Fahrenheit's scale, estimated above 212 or 
simple atmospheric pressure. 

The committee of the Franklin Institute adopted the exponent 
6, and found it necessary to change the constant 0*0029 into 
0*00333 ; thus modified, they represented their experiments by 
the equation 

p = (0*460467 + 0*00521478 tf , . . , (0). 

By combining Dr. Dalton's experiments with the mean between 
those of the French Academy and the Franklin Institute, we obtain 
the following equations, the one being applicable for temperatures 
below 212 degrees, and the other for temperatures above that 
point as far as 50 atmospheres. Thus, for low pressure steam, 
that is, for steam of less temperature than 212, it is 

t + 175x7-71307 

~S8T~^ . . . . (P): 
and for steam above the temperature of 212, it is 



/=C 



/=C 



t -^121y.. 

333 / . . • • W- 



In consequence therefore of the high and imposing authority 
from which these formulas are deduced, we shall adopt them in all 
our subsequent calculations relative to the steam engine ; and in 
order to render their application easy and familiar, we shall trans- 
late them into rules in words at length, and illustrate them by the 
resolution of appropriate numerical examples; and for the sake of 
a systematic arrangement, we think proper to branch the subject 
into a series of problems, as follows : 

The temperature of steam being given in degrees of Fahrenheit's 
thermometer, to find the corresponding elastic force in inches of 
mercury. — The problem, as here propounded, is resolved by one or 
other of the last two equations, and the process indicated by the 
arrangement is thus expressed : — 

Rule. — To the given temperature expressed in degrees of 
Fahrenheit's thermometer, add the constant temperature 175 ; find 
the logarithm answering to the sum, from which subtract the con- 
stant 2*587711 ; multiply the remainder by the index 7*71307, and 
the product will be the logarithm of the elastic force in atmospheres 
of 30 inches of mercury when the given temperature is less than 
212 degrees. But when the temperature is greater than 212, 
increase it by 121 ; then, from the logarithm of the temperature 
thus increased, subtract the constant logarithm 2*522444, multiply 
the remainder by the exponent 6*42, and the product will be the 



196 THE PRACTICAL MODEL CALCULATOR. 

logarithm of the elastic force in atmospheres of 30 inches of mer- 
cury ; which being multiplied by 30 will give the force in inches, 
or if multiplied by 14-76 the result will be expressed in pounds 
avoirdupois per square inch. 

When steam is generated under a temperature of 187 degrees of 
Pahrenheit's thermometer, what is its corresponding elastic force in 
atmospheres of 30 inches of mercury ? 

In this example, the given temperature is less than 212 degrees : 
it will therefore be resolved by the first clause of the preceding 
rule, in which the additive constant is 175 ; hence we get 

187 + 175 = 362. ..log. 2-558709 
Constant divisor = 387... log. 2-587711 subtract 

9-970998 X 7*71307 = 9-773393 

And the corresponding natural number is 0-5934 atmospheres, or 
17*802 inches of mercury, the elastic force required, or if expressed 
in pounds per square inch, it is 0*5934 x 14-76 = 8-76 lbs. very 
nearly. If the temperature be 250 degrees of Fahrenheit, the pro- 
cess is as follows : 

250 + 121 = 371. ..log. 2-569374 
Constant divisor = 333. ..log. 2-522444 subtract 

0-046930 X 6-42 = 0-301291 

And the corresponding natural number is 2-0012 atmospheres, or 
60-036 inches of mercury, and in pounds per square inch it is 
2-0012 X 14-76 = 29-54 lbs. very nearly. 

It is sometimes convenient to express the results in inches of 
mercury, without a previous determination in atmospheres, and for 
this purpose the rule is simply as follows : 

Rule. — Multiply the given temperature in degrees of Fahren- 
heit's thermometer by the constant coefficient 1-5542, and to the 
product add the constant number 271*985; then from the loga- 
rithm of the sum subtract the constant logarithm 2-587711, and 
multiply the remainder by the exponent 7*71307 ; the natural num- 
ber answering to the product, considered as a logarithm, will give 
the elastic force in inches of mercury. This answers to the case 
when the temperature is less than 212 degrees; but when it is 
above that point proceed as follows : 

Multiply the given temperature in degrees of Fahrenheit's ther- 
mometer by the constant coefficient 1*69856, and to the product add 
the constant number 205*526 ; then from the logarithm of the sum 
subtract the constant logarithm 2-522444, and multiply the re- 
mainder by the exponent 6-42 ; the natural number answering to 
the product considered as a logarithm, will give the elastic force 
in inches of mercury. Take, for example, the temperatures as 
assumed above, and the process, according to the rule, is as fol- 
lows ; 



THE STEAM ENGINE. 197 

187 X 1-5542 = 290-6354 
Constant = 271-985 a dd 

Sum = 562-6204. ..log. 2-750216 
Constant = 387 log. 2-587711 subtract 

0-162505 X 7-71307 = 1-253408 
And the natural number answering to this logarithm is 17*923 inches 
of mercury. By the preceding calculation the result is 17*802; 
the slight difference arises from the introduction of the decimal con- 
stants, which in consequence of not terminating at the proper place 
are taken to the nearest unit in the last figure, but the process is 
equally true notwithstanding. For the higher temperature, we get 
250 X 1-69856 = 424-640 

Constant = 205-526 add 

Sum = 630.166 log. 2-799456 

Constant = 333 log. 2-522444 subtract 

0-277011 X 6-42 = 1-778410 

And the natural number answering to this logarithm is 60-036 
inches of mercury, agreeing exactly with the result obtained as 
above. 

It is moreover sometimes convenient to express the force of the 
steam in pounds per square inch, without a previous determination 
in atmospheres or inches of mercury; and when the equations are 
modified for that purpose, they supply us with the following process, 
viz.: 

Multiply the given temperature by the constant coefficient 
1-41666, and to the product add the constant number 247-9155; 
then, from the logarithm of the sum subtract the constant logarithm 
2-587711, and multiply the remainder by the index 7-71307 ; the 
natural number answering to the product will give the pressure in 
pounds per square inch, when the temperature is less than 212 de- 
grees ; but for all greater temperatures the process is as follows : 

Multiply the given temperature by the constant coefficient 
1-5209, and to the product add the constant number 184*0289 ; 
then, from the logarithm of the sum subtract the constant logarithm 
2-522444, and multiply the remainder by the exponent 6-42; the 
natural or common number answering to the product, will express 
the force of the steam in pounds per square inch. If any of these 
results be multiplied by the decimal 0*7854, the product will be the 
corresponding pressure in pounds per circular inch. Taking, there- 
fore, the temperatures previously employed, the operation is as 
follows : 

187 X 1-41666 = 264*9155 

Constant = 247-9155 add 

Sum = 512.8310.log. 2-709974 
Constant = 387 log. 2-587711 subtract 

0-122263 X 7-71307 = 0-942656 

e2 



198 THE PRACTICAL MODEL CALCULATOR. 

And the number answering to this logarithm is 8-763 lbs. per square 
inch, and 8*763 X 0*7854 = 6*8824 lbs. per circular inch, the pro- 
portion in the two cases being as 1 to 0*7554. Again, for the 
higher temperature, it is 

250 X 1*5209 = 380*2250 

Constant = 184*0289 add . 

Sum = 564*2539 log. 2*751475 

Constant = 333 log. 2*522444 subtract 

0*229031 X 6*42 = 1*470279 

And the number answering to this logarithm is 29*568 lbs. per 
square inch, or 29568 X 0*7854 = 23*2226 lbs. per circular inch. 

We have now to reverse the process, and determine the tempera- 
ture corresponding to any given power of the steam, and for this 
purpose we must so transpose the formulas (P) and (Q), as to express 
the temperature in terms of the elastic force, combined with given 
constant numbers ; but as it is probable that many of our readers 
would prefer to see the theorems from which the rules are deduced, 
we here subjoin them. 

For the lower temperature, or that which does not exceed the 
temperature of boiling water, we get 

^^ 249/"^- 175 .... (R). 
Where t denotes the temperature in degrees of Fahrenheit's ther- 
mometer, and /the elastic force in inches of mercury, less than 30 
inches, or one atmosphere ; but when the elastic force is greater 
than one atmosphere, the formula for the corresponding temperature 
is as follows : 

, ^ = 196/^"^- 121 ... . (S). 

In the construction of these formulas, we have, for the sake of 
simplicity, omitted the fractions that obtain in the coefficient of/; 
for since they are very small, the omission will not produce an error 
of any consequence ; indeed, no error will arise on this account, as 
we retain the correct logarithms, a circumstance that enables the 
computer to ascertain the true value of the coefficients whenever it 
is necessary so to do ; but in all cases of actual practice, the results 
derived from the integral coefficients will be quite sufficient. The 
rule supplied by the equations (R) and (S) is thus expressed : 

When the elastic force is less than the pressure of the atmosphere, 
that is, less than 30 inches of the mercurial column, — 

Rule. — Divide the logarithm of the given elastic force in inches 
of mercury, by the constant index 7*71307, and to the quotient add 
the constant logarithm 2*396204; then from the common or natural 
number answering to the sum, subtract the constant temperature 
175 degrees, and the remainder will be the temperature sought in 
degrees of Fahrenheit's thermometer. But when the elastic force 
exceeds 30 inches, or one atmosphere, the following rule applies : 



THE STEAM ENGINE. 199 

Divide the logarithm of the given elastic force in inches of mer- 
cury by the constant index 642, and to the quotient add the con- 
stant logarithm 2-292363 : then, from the natural number answer- 
ing to the sum subtract the constant temperature 121 degrees, and 
the remainder will be the temperature sought. Similar rules might 
be constructed for determining the temperature, when the pressure 
in pounds per square inch is given ; but since this is a less useful 
case of the problem, we have thought proper to omit it. We there- 
fore proceed to exemplify the above rules, and for this purpose we 
shall suppose the pressure in the two cases to be equivalent to the 
weight of 19 and 60 inches of mercury respectively. The operations 
will therefore be as follows : 

Log. 19 -- 7-71307 = 1-278754 -^ 7-71307 = 0-165791 
Constant coefficient = 249 log. 2-396204 add 

Natural number = 364-75 log. 2-561994 

Constant temperature = 175 subtract 

Required temperature = 189-75 degrees of Fahrenheit's scale. 
For the higher elastic force the operation is as follows : 

Log. 60 ^ 6-42 = 1-778151 -r- 6-42 = 0-276969 
Constant coefficient = 196 log. 2-292363 add 

Natural number = 370-97 log. 2-569332 

Constant temperature = 121 subtract 

Required temperature = 249-97 degrees of Fahrenheit's scale. 

All the preceding results, as computed by our rules, agree as 
nearly with observation as can be desired : but they have all been 
obtained on the supposition that the steam is in contact with the 
liquid from which it is generated ; and in this case it is evident 
that the steam must always attain an elastic force corresponding to 
the temperature ; and in accordance to any increase of pressure, 
supposing the temperature to remain the same, a quantity of it 
corresponding to the degree of compression must simply be condensed 
into water, and in consequence will leave the diminished space 
occupied by steam of the original degree of tension ; or otherwise 
to express it, if the temperature and pressure invariably correspond 
with each other, it is impossible to increase the density and elas- 
ticity of the steam except by increasing the temperature at the same 
time ; and, contrariwise, the temperature cannot be increased with- 
out at the same time increasing the elasticity and density. This 
being admitted, it is obvious that under these circumstances the 
steam must always maintain its maximum of pressure and density : 
but if it be separated from the liquid that produces it, and if its 
temperature in this case be increased, it will be found not to possess 
a higher degree of elasticity than a volume of atmospheric air simi- 
larly confined, and heated to the same temperature. Under this 
new condition, the state of maximum density and elasticity ceases ; 
for it is obvious that since no water is present, there cannot be any 






200 THE PRACTICAL MODEL CALCULATOR. 

more steam generated by an increase of temperature : and conse- 
quently the force of the steam is only that which confines it to its 
original bulk, and is measured by the effort which it exerts to ex- 
pand itself. Our next object, therefore, is to inquire what is the 
law of elasticity of steam under the conditions that we have here 
specified. 

The specific gravity of steam, its density, and the volume which 
it occupies at different temperatures, have been determined by ex- 
periment with very great precision ; and it has also been ascertained 
that the expansion of vapour by means of heat is regulated by the 
same laws as the expansion of the other gases, viz. that all gases 
expand from unity to 1-375 in bulk by 180 degrees of temperature; 
and again, that steam obeys the law discovered by Boyle and Mari- 
otte, contracting in volume in proportion to the degree of pressure 
which it sustains. We have therefore to inquire what space a given 
quantity of water converted into steam will occupy at a given pres- 
sure ; and from thence we can ascertain the specific gravity, density, 
and volume at all other pressures. 

When a gas or vapour is submitted to a constant pressure, the 
quantity which it expands by a given rise of temperature is calcu- 
lated by the following theorem, 

't' -f 459x ^rp. 

^4- 4597 ^ ^ 

where t and t' are the temperatures, and v, v' the corresponding 
volumes before and after expansion; hence this rule. 

Rule. — To each of the temperatures before and after expansion, 
add the constant experimental number 459 ; divide the greater sum 
by the lesser, and multiply the quotient by the volume at the lower 
temperature, and the product will give the expanded volume. 

If the volume of steam at the temperature of 212 degrees of Fah- 
renheit be 1711 times the bulk of the water that produces it, what 
will be its volume at the temperature of 250-3 degrees, supposing 
the pressure to be the same in both cases ? 

Here, by the rule, we have 212 + 459 = 671, and 250-3 -f 459 
= 709-3 ; consequently, by dividing the greater by the lesser, and 

multiplying by the given volume, we get ~ — x 1711 = 1808-66 

for the volume at the temperature of 250*3 degrees. 

Again, if the elastic force at the lower temperature and the cor- 
responding volume be given, the elastic force at the higher tem- 
perature can readily be found ; for it is simply as the volume the 
vapour occupies at the lower temperature is to the volume at the 
higher temperature, or what it would become by expansion, so is the 
elastic force given to that required. 

If the volume which steam occupies under any given pressure 
and temperature be given, the volume which it will occupy under 
any proposed pressure can readily be found by reversing the pre- 
ceding process, or by referring to chemical tables containing the 



THE STEAM ENGINE. 201 

specific gravity of the gases compared with air as unity at the same 
pressure and temperature. Now, air at the mean state of the at- 
mosphere has a specific gravity of If as compared with water at 
1000 ; and the bulks are inversely as the specific gravities, accord- 
ing to the general laws of the properties of matter previously an- 
nounced ; hence it follows that air is 818 times the bulk of an 
equal weight of water, for 1000 -^ If = 818-18. But, by the 
experiments of Dr. Dalton, it has been found that steam of the 
same pressure and temperature has a specific gravity of '625 com- 
pared with air as unity ; consequently, we have only to divide the 
number 818*18 by '625, and the quotient will give the propor- 
tion of volume of the vapour to one of the liquid from which it is 
generated ; thus we get 81848 -^ '625 = 1309 ; that is, the volume 
of steam at 60 degrees of Fahrenheit, its force being 30 inches of 
mercury, is 1309 times the volume of an equal weight of water ; 
hence it follows, from equation (T), that when the temperature in- 
creases to t^, the volume becomes 

/459 + f \ 
v^ = 1309 X ( 459 ^ 6q ) = 2-524(459 + f); 

and from this expression, the volume corresponding to any specified 
elastic force /, and temperature t^, may easily be found ; for it is 
inversely as the compressing force : that is, 

/: 30: : 2-525(459 -f f) : v' ; 
consequently, by working out the analogy, w^e get 
^ _ 75-67(459 + t' ). ^^j_ 

By this theorem is found the volume of steam as compared with 
that of the water producing it, when under a pressure correspond- 
ing to the temperature. The rule in words is as follows : 

Rule. — Calculate the elastic force in inches of mercury by the 
rule already given for that purpose, and reserve it for a divisor. 
To the given temperature add the constant number 459, and mul- 
tiply the sum by 75-67 ; then divide the product by the reserved 
divisor, and the quotient will give the volume sought. 

When the temperature of steam is 250-3 degrees of Fahrenheit's 
thermometer, what is the volume, compared with that of water ? 

The temperature being greater than 212 degrees, the force is cal- 
culated by the rule to equation (Q), and the process is as follows : 

250-3 -f 121 = 371-3 log. 2-5697249 
Constant divisor = 333 log. 2-52 24442 subtract 

0-0472807 x6-42=0-3035421 
Atmosphere = 30 inches of mercury log. 1-4771213 add 

Elastic force = 60-348 log. 1-7806634 "| 

Again it is, [ , 

459 -f 250-3 = 709-3 log. 2-8508300 \ ,, r"^^' 

Constantcoefficient = 75-67 log. 1-8789237 j ^^ 4-7297537 J 
Volume = 889-39 times that of water, log. 2-9490903 re- 
mainder. 



202 THE PRACTICAL MODEL CALCULATOR. 

Thus we have given the method of calculating the elastic force 
of steam when the temperature is given either in atmospheres or 
inches of mercury, and also in pounds or the square or circular 
inch : we have also reversed the process, and determined the tem- 
perature corresponding to any given elastic force. We have, 
moreover, shown how to find the volume corresponding to different 
temperatures, when the pressure is constant ; and, finally, we have 
calculated the volume, when under a pressure due to the elastic 
force. These are the chief subjects of calculation as regards the 
properties of steam ; and we earnestly advise our readers to render 
themselves familiar with the several operations. The calculations 
as regards the motion of steam in the parts of an engine to produce 
power, will be considered in another part of the present treatise. 

The equation (U), we may add, can be exhibited in a different 
form involving only the temperature and known quantities ; for 
since the expressions (P) and (Q) represent the elastic force in terms 
of the temperature, according as it is under or above 212 degrees 
of Fahrenheit, we have only to substitute those values of the elastic 
force when reduced to inches of mercury, instead of the symbol/ 
in equation (U), and we obtain, when the temperature is less than 
212 degrees, 

Vol.=75-67(tem.+459)H-(-004016xtem. + -7028077-^^^ . (V). 

and when the temperature exceeds 212 degrees, the expression be- 
comes 

Vol.=75-67(tem.+459)---005101xtem. + -617195f^ . (W.) 

These expressions are simple in their form, and easily reduced ; 
but, in pursuance of the plan we have adopted, it becomes necessary 
to express the manner of their reduction in words at length, as 
follows : 

Rule. — When the given temperature is under 212 degrees, mul- 
tiply the temperature in degrees of Fahrenheit's thermometer by 
the constant fraction -004016, and to the product add the constant 
increment -702807 ; multiply the logarithm of the sum by the in- 
dex 7-71307, and find the natural or common number answering to 
the product, which reserve for a divisor. To the temperature add 
the constant number 459, and multiply the sum by the coefficient 
75-67 for a dividend ; divide the latter result by the former, and 
the quotient will express the volume of steam when that of water is 
unity. 

Again, when the given temperature is greater than 212 degrees, 
multiply it by the fraction -005101, and to the product add the 
constant increment -617195 ; multiply the logarithm of the sum 
by the index 6-42, and reserve the natural number answering to 
the product for a divisor ; find the dividend as directed above, 
which, being divided by the divisor, will give the volume of steam 
when that of the water is unity. 

How many cubic feet of steam will be supplied by one cubic foot 



THE STEAM ENGINE. 203 

of water, under the respective temperatures of 187 and 293-4 de- 
grees of Fahrenheit's thermometer ? 
Here, by the rule, we have 
187x0-004016=0-750992 
Constant increment=0 -702807 

Sum =1-453799 log. -1625043 x 7-71307=l-2534069 
and the number answering to this logarithm is 17*92284, the di- 
visor. But 187 + 459 = 646, and 646 x 75-67 = 48882-82, the 
dividend ; hence, by division, we get 48882-82 -J- 17-92284 = 
2727*4 cubic feet of steam from one cubic foot of water. 
Again, for the higher temperature, it is 

293-4x0-005101 = 1-496633 
Constant increment = 0-617195 

Sum = 2413828 log. 0-3250696x642=2-0869468; 
and the number answering to this logarithm is 122-165, the divisor. 
But 293-4 + 459 = 752-4, and 752-4 x 75-67 = 56934-108, the 
dividend ; therefore, by division, we get 56934-108 h- 122-165 = 
466-04 cubic feet of steam from one cubic foot of water. 

The preceding is a very simple process for calculating the volume 
which the steam of a cubic foot of water will occupy when under 
a pressure due to a given temperature and elastic force ; and since 
a knowledge of this particular is of the utmost importance in cal- 
culations connected with the steam engine, it is presumed that our 
readers will find it to their advantage to render themselves familiar 
with the method of obtaining it. The above example includes both 
cases of the problem, a circumstance which gives to the operation, 
considered as a whole, a somewhat formidable appearance : but it 
would be difficult to conceive a case in actual practice where the 
application of both the formulas will be required at one and the 
same time ; the entire process must therefore be considered as em- 
bracing only one of the cases above exemplified ; and consequently 
it can be performed with the greatest facility by every person who 
is acquainted with the use of logarithms ; and those unacquainted 
with the application of logarithms ought to make themselves masters 
of that very simple mode of computation. 

Another thing which it is necessary sometimes to discover in 
reasoning on the properties of steam as referred to its action in a 
steam engine, is the weight of a cubic foot, or any other quantity 
of it, expressed in grains, corresponding to a given temperature and 
pressure. Now, it has been ascertained by experiment, that when 
the temperature of steam is 60 degrees of Fahrenheit, and the 
pressure equal to 30 inches of mercury, the weight of a cubic foot 
in grains is 329-4 ; but the weight is directly proportional to the 
elastic force, for the elastic force is proportional to the density : 
consequently, if/ denote any other elastic force, and lo the weight 
in grains corresponding thereto, then we have 

30 :/:: 329*4 ; «(; = 10-98/, 



204 THE PRACTICAL MODEL CALCULATOE. 

the weiglit of a cubic foot of vapour at the force /, and temperature 
60 degrees of Fahrenheit. Let t denote the temperature at the 

459 + t 459 + < 
force/; then by equation (T), we have v = .rq , ^q = — ^Tq~> 

the volume at the temperature t, supposing the volume at 60 de- 
grees to be unity ; that is, one cubic foot. Now, since the den- 
sities are inversely proportional to the spaces which the vapour oc- 

(459 + t} ^ , 519z^ ^ ^ ^ 

cupies, we nave — rrn — : 1 : : zv : w = .rq , . ; but by the 

preceding analogy, the value of w is 10*98/; therefore, by substi- 
tution, we get 

, 5698-62/ 
-'=459Ti • • • • (^)- 

This equation expresses the weight in grains of a cubic foot of 
steam at the temperature t and force/; and if we substitute the 
value of /, from equations (P) and (Q), reduced to inches of mer- 
cury, and modified for the two cases of temperature below and 
above 212 degrees of Fahrenheit, we shall obtain, in the first case, 

w' = (0-012324 X temp, -f 2-155611)^-^^307^ (^emp. -f 459). . . .(Y) 

and for the second case, where the temperature exceeds 212, it is 

w' = (0-01962 X temp. + 2-37374)^-^ h- (temp, -f 459) . . . (Z) 

These two equations, like those marked (Y) and (W) are suf- 
ficiently simple in their form, and ofi"er but little difficulty in their 
application. The rule for their reduction when expressed in words 
at length, is as follows : 

Rule. — When the temperature is less than 212 degrees, multi- 
ply the given temperature, in degrees of Fahrenheit's thermometer, 
by the fraction 0-012324, and to the product add the constant in- 
crement 2-155611 ; then multiply the logarithm of the sum by the 
index 7-71307, and from the product subtract the logarithm of the 
temperature, increased by 459 ; the natural number answering to 
the remainder will be the weight of a cubic foot in grains. 

Again, when the temperature exceeds 212, multiply it by the 
fraction 0-01962, and to the product add the constant increment 
2-37374 ; then multiply the logarithm of the sum by the index 6-42, 
and from the product subtract the logarithm of the temperature in- 
creased by 459 ; the natural number answering to the remainder 
will be the weight of a cubic foot in grains. 

Supposing the temperatures to be as in the preceding example, 
what will be the weight of a cubic foot in grains for the two cases ? 

Here, by the rule, we have 

187 X 0-012324 = 2-304588 
Constant increment = 2-155611 

Sum = 4-460199 log. 0-6493542 x 7-71307 = 5-0085143 
187 4- 459 = 646 log. 2-8102325, subtract 

Natural number = 157-863 grains per cubic foot log. 2-1982818 



THE STEAM ENGINE. 205 



For the higher temperature, it is 



293-4 X 0-01962 = 5-756508 
Constant increment = 2-373740 



Sum = 8-130248 log. 0-9101088 X 6-42 = 5-8428664 
293-4 + 459 = 752-4 . . . . log. 2-8764488 , subtract 

Natural number = 925-59 grains per cubic foot . log. 2-9664176 

Here again the operation resolves both cases of the problem ; 

but in practice only one of them can be required. 

THE MOTION OF ELASTIC FLUIDS. 

The next subject that claims our attention is the velocity with 
which elastic fluids or vapours move in pipes or confined passages. 
It is a well-known fact in the doctrine of pneumatics, that the mo- 
tion of free elastic fluids depends upon the temperature and pres- 
sure of the atmosphere ; and, consequently, when an elastic fluid 
is confined in a close vessel, it must be similarly circumstanced 
with regard to temperature and pressure as it would be in an at- 
mosphere competent to exert the same pressure upon it. The sim- 
plest and most convenient way of estimating the motion of an elastic 
fluid is to assign the height of a column of uniform density, capable 
of producing the same pressure as that which the fluid sustains in 
its state of confinement ; for under the pressure of such a column, 
the velocity into a perfect vacuum will be the same as that acquired 
by a heavy body in falling through the height of the homogeneous 
column, a proper allowance being made for the contraction at the 
aperture or orifice through which the fluid flows. 

When a passage is opened between two vessels containing fluids 
of difi'erent densities, the fluid of greatest density rushes out of the 
vessel that contains it, into the one containing the rarer fluid, and 
the velocity of influx at the first instant of the motion is equal to 
that which a heavy body acquires in falling through a certain 
height, and that height is equal to the difi'erence of two uniform 
columns of the fluid of greatest density, competent to produce the 
pressures under which the fluids are originally confined ; and the 
velocity of motion at any other instant is proportional to the square 
root of the difi'erence between the heights of the uniform columns 
producing the pressures at that instant. Hence we infer that the 
velocity of motion continually decreases,— the density of the fluids 
in the two vessels approaching nearer and nearer to an equality, 
and after a certain time an equilibrium obtains, and the velocity 
of motion ceases. 

It is abundantly confirmed by observation and experiment, that 
oblique action produces very nearly the same effect in the motion 
of elastic fluids through apertures as it does in the case of water ; 
and it has moreover been ascertained that eddies take place under 
similar circumstances, and these eddies must of course have a ten- 
dency to retard the motion : it therefore becomes necessary, in all 
the calculations of practice, to make some allowance for the retard- 
ation that takes place in passing the orifice ; and this end is most 
s 



206 THE PRACTICAL MODEL CALCULATOR. 

conveniently answered by modifying the constant coefficient ac- 
cording to the nature of the aperture through which the motion 
is made. Numerous experiments have been made to ascertain the 
effect of contraction in orifices of different forms and under dif- 
ferent conditions, and amongst those which have proved the most 
successful in this respect, we may mention the experiments of Du 
Buat and Eytelwein, the latter of whom has supplied us with a 
series of coefficients, which, although not exclusively applicable to 
the case of the steam engine, yet, on account of their extensive 
utility, we take the liberty to transcribe. They are as follow : — 

1. For the velocity of motion that would re- 
sult from the direct unretarded action of 

the column of the fluid that produces it, we 

have 3 Y= N/579A 

2. For an orifice or tube in the form of the 

contracted vein 10 V = ^/6084^ 

3. For wide openings having the sill on a^ 

level with the bottom of the reservoir ... | 

4. For sluices with walls in a line with the J> 10 V = -v/5929A 
orifice 

5. For bridges with pointed piers J 

6. For narrow openings having the sill on a^ 
level with the bottom of the reservoir ... 

7. For small openings in a sluice with side 
walls 

8. For abrupt projections 

9. For bridges with square piers 

10. For openings in sluices without side walls 10 V = ^/2601/i 

11. For openings or orifices in a thin plate V = \/25A 

12. For a straight tube from 2 to 3 diameters 

in length projecting outwards 10 V = v/4225 

13. For a tube from 2 to 3 diameters in length 

projecting inwards 10 V = v/2976-25A 

It is necessary to observe, that in all these equations Y is the 
velocity of motion in feet per second, and A the height of the co- 
lumn producing it, estimated also in feet. Nos. 1, 2, 11, 12, and 
13 are those which more particularly apply to the usual passages 
for the steam in a steam engine ; but since all the others meet their 
application in the every-day practice of the civil engineer, we have 
thought it useful to supply them. 

MOTION or STEAM IN AN ENGINE. 

We have already stated that the best method of estimating the 
motion of an elastic fluid, such as steam or the vapour of water, is 
to assign the height of a uniform column of that fluid capable of 
producing the pressure : the determination of this column is there- 
fore the leading step of the inquiry ; and since the elastic force of 
steam is usually reckoned in inches of mercury, 30 inches being 



10 V = ^/4761^ 



THE STEAM ENGINE. 207 

equal to the pressure of the atmosphere, the subject presents but 
little diiBculty ; for we have already seen that the height of a co- 
lumn of water of the temperature of 60 degrees, balancing a column 
of 30 inches of mercury, is 34*023 feet ; the corresponding column 
of steam must therefore be as its relative bulk and elastic force ; 
hence we have 30 : 34*023 : f v. h = 1-1341 /v, where / is the 
elastic force of the steam in inches of mercury, v the correspond- 
ing volume or bulk when that of water is unity, and A the height 
of a uniform column of the fluid capable of producing the pressure 
due to the elastic force ; consequently, in the case of a direct un- 
retarded action, the velocity into a perfect vacuum, according to 
No. 1 of the preceding class of formulas, is V = 8-542 -v// v ; but 
for the best form of pipes, or a conical tube in form of the con- 
tracted vein, the velocity into a vacuum, according to No. 2, be- 
comes V = 8-307 Vf V ; and for pipes of the usual construction, 
No. 12 gives V = 6-922 ^//^; No. 13 gives V = 5-804 v//V; 
and in the case of a simple orifice in a thin plate, we get from 
No. 11 V = 5-322 's/f V. The consideration of all these equa- 
tions may occasionally be required, but our researches will at pre- 
sent be limited to that arising from No. 12, as being the best 
adapted for general practice ; and for the purpose of shortening 
the investigation, we shall take no further notice of the case in 
which the temperature of the steam is below 212 degrees of Fah- 
renheit ; for the expression which indicates the velocity into a va- 
cuum being independent of the elastic force, a separate considera- 
tion for the two cases is here unnecessary. 

It has been shown in the equation marked (U), that the volume 
of steam which is generated from an unit of water, i^ v = 

75-67 (temp. + 459) ^ , . ^ ^ ^ ^ . , , . . 
^ — ^ ; let this value oi v be substituted for it in 

the equation Y = 6*922 \/f v, and we obtain for the velocity into 
a vacuum for the usual form of steam passages, as follows, viz. : 

V = 60-2143 v/(temp. + 459)". 
This is a very neat and simple expression, and the object de- 
termined by it is a very important one : it therefore merits the 
reader's utmost attention, especially if he is desirous of becoming 
familiar with the calculations in reference to the motion of steam. 
The rule which the equation supplies, when expressed in words at 
length, is as follows : — 

Rule. — To the temperature of the steam, in degrees of Fahren- 
heit's thermometer, add the constant number or increment 459, and 
multiply the square root of the sum by 60-2143 ; the product will 
be the velocity with which the steam rushes into a vacuum in feet 
per second. 

With what velocity will steam of 293*4 degrees of Fahrenheit's 
thermometer rush into a vacuum when under a pressure due to the 
elastic force corresponding to the given temperature. 



208 THE PRACTICAL MODEL CALCULATOR. 

By the rule it is 293-4 + 459 = 752-4 i log. 1-4382244 

Constant coefficient = 60-2143 log. 1-7797018 add 

Velocity into a vacuum in feet per second = 1651-68 log. 3-2179262 

This is the velocity into a perfect vacuum, when the motion is 
made through a straight pipe of uniform diameter ; but when the 
pipe is alternately enlarged and contracted, the velocity must ne- 
cessarily be reduced in proportion to the nature of the contraction ; 
and it is further manifest, that every bend and angle in a pipe will 
be attended with a correspondent diminution in the velocity of mo- 
tion : it therefore behoves us, in the actual construction of steam 
passages, to avoid these causes of loss as much as possible ; and 
where they cannot be avoided altogether, such forms should be 
adopted as will produce the smallest possible retarding effect. In 
cases where the forms are limited by the situation and conditions 
of construction, such corrections should be applied as the circum- 
stances of the case demand ; and the amount of these corrections 
must be estimated according to the nature of the obstructions them- 
selves. For each right-angled bend, the diminution of velocity is 
usually set down as being about one-tenth of its unobstructed value ; 
but whether this conclusion be correct or not, it is at least certain 
that the obstruction in the case of a right-angled bend is much 
greater than in that of a gradually curved one. It is a very com- 
mon thing, especially in steam vessels, for. the main steam pipe to 
send off branches at right angles to each cylinder, and it is easy to 
see that a great diminution in the velocity of the steam must take 
place here. In the expansion valve chest a further obstruction 
mu^t be met with, probably to the extent of reducing the velocity 
of the steam two-tenths of its whole amount. 

These proportional corrections are not to be taken as the results 
of experiments that have been performed for the purpose of deter- 
mining the effect of 'the above causes of retardation: we have no 
experiments of this sort on which reliance can be placed ; and, in 
consequence, such elements can only be inferred from a comparison 
of the principles that regulate the motion of other fluids under simi- 
lar circumstances : they will, however, greatly assist the engineer 
in arriving at an approximate estimate of the diminution that takes 
place in the velocity in passing any number of obstructions, when 
the precise nature of those obstructions can be ascertained. In the 
generality of practical cases, if the constant coefficient 60*2143 be 
reduced in the ratio of 650 to 450, the resulting constant 41-6868 
may be employed without introducing an error of any consequence. 

OF THE ASCENT OF SMOKE AND HEATED AIR IN CHIMNEYS. 

The subject of chimney flues, with the ascent of smoke and heated 
air, is another case of the motion of elastic fluids, in which, by a 
change of temperature, an atmospheric column assumes a different 
density from another, where no such alteration of temperature oc- 
curs. The proper construction of chimneys is a matter of very 
great importance to the practical engineer, for in a close fireplace, 



THE STEAM ENGINE. 209 

designed for the generation of steam, there must be a considerable 
draught to accomplish the intended purpose, and this depends upon 
the three following particulars, viz. : 

1. The height of the chimney from the throat to the top. 

2. The area of the transverse section. 

3. The temperature at which the smoke and heated air are al- 
lowed to enter it. 

The formula for determining the power of the chimney may be 
investigated in the following manner : 

Put h = the height in feet from the place where the flue enters 

to the top of the chimney, 
h = the number of cubic feet of air of atmospheric density 

that the chimney must discharge per hour, 
a = the area of the aperture in square inches through which 

h cubic feet of air must pass when expanded by a 

change of temperature, 
V = the velocity of ascent in feet per second, 
f = the temperature of the external air, and 
t = the temperature of the air to be discharged by the 

chimney. 

Now the force producing the motion in this case is manifestly 
the difference between the weight of a column of the atmospheric 
air and another of the air discharged by the chimney : and when 
the temperature of the atmospheric air is at 52 degrees of Fahren- 
heit's thermometer, this difference will be indicated by the term 

( tf — t \ 
h W/ I 4;rQ7 ; the velocity of ascent will therefore be 

V — J 64f h ^ , , .rq > feet per second, and the quantity of air 

discharged per second will therefore be, <^ J ^^ t "^ ./ ■ ^rQ ^ , 

supposing that there is no contraction in the stream of air ; but it 
is found by experiment, that in all cases the contraction that takes 
place diminishes the quantity discharged, by about three-eighths of 
the whole ; consequently, the quantity discharged per hour in cu- 
bic feet becomes 



b = 125-69 a. ' '" '" ^^ 



•m 



459* 

This would be the quantity discharged, provided there were no 
increase of volume in consequence of the change of temperature ; 

but air expands from h to , .rq for t' — t degrees of tem- 

perature, as has been shown elsewhere ; consequently, by compa- 
rison, we have 



= 125-69 



t + 459 - '■''•' """>!{ + 459 
e2 u 



n 1^ (*' 



210 THE PRACTICAL MODEL CALCULATOR. 

From this equation, therefore, any one of the quantities "which 
it involves can be found, when the others are given : it however 
supposes that there is no other cause of diminution but the contrac- 
tion at the aperture ; but this can seldom if ever be the case ; for 
eddies, loss of heat, obstructions, and change of direction in the 
chimney, will diminish the velocity, and consequently a larger area 
will be required to suffer the heated air to pass. A sufficient al- 
lowance for these causes of retardation will be made, if we change 
the coefficient 125-69 to 100 ; and in this case the equation for the 
area of section becomes 



a = h V{t' + 459)^ -^ 100 it + 459) Vh {f - t). 

And if ~we take the mean temperature of the air of the atmo- 
sphere at 52 degrees of Fahrenheit, and make an allowance of 16 
degrees for the difference of density between atmospheric air and 
coal smoke, our equation will ultimately assume the form 



a = h Vit' + 459)^ -- 51100 Vh {i! - t - 16). 

It has been found by experiment that 200 cubic feet of air of at- 
mospheric density are required for the complete combustion of one 
pound of coal, and the consumption of ten pounds of coal per hour 
is usually reckoned equivalent to one horse power : it therefore ap- 
pears that 2000 cubic feet of air per hour must pass through the fire 
for each horse power of the engine. This is a large allowance, but 
it is the safest plan to calculate in excess in the first instance ; for 
the chimney may afterwards be convenient, even if considerably 
larger than is necessary. The rule for reducing the equation is as 
follows : — 

Rule. — Multiply the number of horse power of the engine by 
the f power of the temperature at which the air enters the chimney, 
increased by 459; then divide the product by 25*55 times the 
square root of the height of the chimney in feet, multiplied by the 
difference of temperature, less 16 degrees, and the quotient will be 
the area of the chimney in square inches. 

Suppose the height of the chimney for a 40-horse engine to be 
70 feet, what should be its area when the difference between the 
temperature at which the air enters the flue, and that of the 
atmosphere is 250 degrees ? 

Here, by the rule, we have, 

250 + 52 = 302, the temperature at which the air enters 
Constant increment = 459 [the flue. 

Sum = 761 log. 2-8813847 

3 

2 )8-6441541 

4-3220770 
Number of horse power = 40 log. 1-6020600 

5-9241370 



THE STEAM ENGHNE. 211 

5-9241370 
250 - 16 = 234 ... . log. 2-3692159 
height = 70 feet . . log. 1-8450980 

2) 4-2143139 

2-1071569 \ 
Constant = 25-55 . . log. 1.4073909 j . . . 3-5145478 

Hence the area of the chimney in square inches is 256-79, log. 
2-4095892; and in this way may the area be calculated for any 
other case ; but particular care must be taken to have the data ac- 
curately determined before the calculation is begun. In the above 
example the particulars are merely assumed ; but even that is suffi- 
cient to show the process of calculation, which is more immediately 
the object of the present inquiry. It is right, however, to add, 
that recent experiments have greatly shaken the doctrine that it is 
beneficial to make chimneys small at the top, though such is the 
way in which they are, nevertheless, still constructed, and our rules 
must have reference to the present practice. It appears, however, 
that it would be the best way to make chimneys expand as they 
ascend, after the manner of a trumpet, with its mouth turned down- 
wards : but these experiments require further confirmation. 

The method of calculation adopted above is founded on the prin- 
ciple of correcting the temperature for the difierence between the 
specific gravity of atmospheric air and that of coal-smoke, the one 
being unity and the other 1*05 ; there is, however, another method, 
somewhat more elegant and legitimate, by employing the specific 
gravity of coal-smoke itself: the investigation is rather tedious and 
prolix, but the resulting formula is by no means difficult ; and since 
both methods give the same result when properly calculated, we 
make no further apology for presenting our readers with another 
rule for obtaining the same object. The formula is as follows : 

= ^ (^' + 459) / 1 ~ 

2757-5 \A(t'- 77-55) 
where a is the area of the transverse section of the chimney in 
square inches, h the quantity of atmospheric air required for com- 
bustion of the coal in cubic feet per hour, h the height of the chim- 
ney in feet, and t' the temperature at which the air enters the flue 
after passing through the fire. The rule for performing this pro- 
cess is thus expressed : 

HuLE. — From the temperature at which the air enters the chim- 
ney, subtract the constant decrement 77-55 ; multiply the remainder 
by the height of the chimney in feet, divide unity by the product, 
and extract the square root of the quotient. To the temperature 
of the heated air, add the constant number 459 ; multiply the sum 
by the number of cubic feet required for combustion per hour, and 
divide the product by the number 2757-5 ; then multiply the quo- 
tient by the square root found as above, and the product will be the 
number of square inches in the transverse section of the chimney. 



a 



212 THE PKACTICAL MODEL CALCULATOR. 

Suppose a mass of fuel in a state of combustion to require 5000 
cubic feet of air per hour, what must be the size of the chimney 
when its height is 100 feet, the temperature at which the heated 
air enters the chimney being 200 degrees of Fahrenheit's ther- 
mometer ? 

By the rule we have 200-77-55=122-45 . . log. 2-0879588 
Height of the chimney =100. . . . log. 2-0000000 

4-0879588 

2)5-9120412 

7-9560206 
200+459=659 . . . log. 2-8188854) 

5000 . . . log. 3-6989700 y add 3-0773399 

2757-5 ar. co. log. 6-5594845 j 

1-0333605 10-798 in. 
This appears to be a very small flue for the quantity of air that 
passes through it per hour ; but it must be observed that we have 
assumed a great height for the shaft, which has the effect of cre- 
ating a very powerful draught, thereby drawing off the heated air 
with great rapidity. 

The advantage of a high flue is so very great, that the reader 
may be desirous of knowing to what height a chimney of a given 
base may be carried with safety, in cases where it is inconvenient 
to secure it with lateral stays ; and, as an approximate rule for this 
purpose is not difficult of investigation, we think proper to supply 
it here. 

When the chimney is equally wide throughout its whole height, 
the formula is 



-Wi 



156 



12000 -ihw; 

but when the side of the base is double the size of the top, the 
equation becomes 



= A^ 



104 



12000 - 0-42 h 



where s is the side of the base in feet, h the height, and m the 
weight of one cubic foot of the material. When the chimney stalk 
is not square, but longer on the one side than the other, s must be 
the least dimension. The proportion of solid wall to a given base, 
as sanctioned by experience, is about two-thirds of its area, conse- 
quently w ought to be two-thirds of the weight of a cubic foot of 
brickwork. Now, a cubic foot of dried brickwork is, on an average, 
114 lbs. ; consequently w = 76 lbs. ; and if this be substituted in 
the foregoing equations, we get for a chimney of equal size through- 
out, 



=^V 



156 



1200 -25 A; 



THE STEAM ENGINE. 213 

and when the chimney tapers to one-half the size at top, it is 



V; 



104 



VdT 



12000 - 32 A; 

"where it may be remarked that 12000 lbs. is the cohesive force of 
one square foot of mortar ; and in the investigation of the formulas 
we have assumed the greatest force of the wind on a square foot 
of surface at 52 lbs. These equations are too simple in their form 
to require elucidation from us ; we therefore leave the reduction as 
an exercise to the reader, who it is presumed will find no difficulty 
in resolving the several cases that may arise in the course of his 
practice. 

2gllafD 
2^K(L + H' 

is the expression given by M. Peclet for the velocity of smoke in 
a chimney, v, the velocity ; t, the temperature, whose maximum 
value is about 300° centigrade ; g = 32^ feet ; D, the diameter 
of the chimney ; H, the height ; L, the length of horizontal flues, 
supposing them formed into a cylinder of the same diameter 
as that of the chimney. K = -0127 for brick, = -005 for sheet- 
iron, and = '0025 for cast-iron chimneys, a = '00365. 

Let L=60; H=150 ; D=5 ; K=-00 5; 2g=6^; t=300° ; 
/ 2gRatI> 
«=-00365. Then v= V D+2^K(H-fL) "" ^^'^^^ feet. 

A cubic foot of water raised into steam is reckoned equivalent to 
a horse power, and to generate the steam with sufficient rapidity, 
an allowance of one square foot of fire-bars, and one square yard 
of efi'ective heating surface, are very commonly made in practice, 
at least in land engines. These proportions, however, greatly vary 
in different cases ; and in some of the best marine engine boilers, 
where the area of fire-grate is restricted by the breadth of the ves- 
sel, and the impossibility of firing long furnaces efi'ectually at sea, 
half a square foot of fire-grate per horse power is a very common 
proportion. Ten cubic feet of water in the boiler per horse power, 
and ten cubic feet of steam room per horse power, have been as- 
signed as the average proportion of these elements ; but the fact is, 
no general rule can be formed upon the subject, for the proportions 
which would be suitable for a wagon boiler would be inapplicable 
to a tubular boiler, whether marine or locomotive ; and good ex- 
amples will in such cases be found a safer guide than rules which 
must often give a false result. A capacity of three cubic feet per 
horse power is a common enough proportion of furnace-room, and 
it is a good plan to make the furnaces of a considerable width, as 
they can then be fired more efi'ectually, and do not produce so much 
smoke as if they are made narrow. As regards the question of 
draft, there is a great diff"erence of opinion among engineers upon 
the subject, some preferring a very sIoav draft and others a rapid 
one. It is obvious that the question of draft is virtually that of 



214 THE PRACTICAL MODEL CALCULATOR. 

the area of fire-grate, or of the quantity of fuel consumed upon a 
given area of grate surface, and the -u'eight of fuel burned on a foot 
of fire-grate per hour varies in difi'erent cases in practice from 3J 
to 80 lbs. Upon the quickness of the draft again hinges the ques- 
tion of the proper thickness of the stratum of incandescent fuel 
upon the grate ; for if the draft be very strong, and the fire at the 
same time be thin, a great deal of uncombined oxygen will escape 
up through the fire, and a needless refrigeration of the contents of 
the flues will be thereby occasioned; whereas, if the fire be thick, 
and the draft be sluggish, much of the useful effect of the coal will 
be lost by the formation of carbonic oxide. The length of the cir- 
cuit made by the smoke varies in almost every boiler, and the same 
may be said of the area of the flue in its cross section, through 
which the smoke has to pass. As an average, about one-fifth of 
the area of fire-grate for the area of the flue behind the bridge, 
diminished to half that amount for the area of the chimney, has 
been given as a good proportion, but the examples which we have 
given, and the average flue area of the boilers which we shall 
describe, may be taken as a safer guide than any such loose state- 
ments. When the flue is too long, or its sectional area is insuffi- 
cient, the draft becomes insufficient to furnish the requisite quantity 
of steam ; whereas if the flue be too short or too large in its area, 
a large quantity of the heat escapes up the chimney, and a depo- 
sition of soot in the flues also takes place. This last fault is one 
of material consequence in the case of tubular boilers consuming 
bituminous coal, though indeed the evil might be remedied by block- 
ing some of the tubes up. The area of water-level is about 5 feet 
per horse power in land boilers. In many cases, however, it is 
much less ; but it is always desirable to make the area of the water- 
level as large as possible, as, when it is contracted, not only is the 
water-level subject to sudden and dangerous fluctuations, but water 
is almost sure to be carried into the cylinder with the steam, in 
consequence of the violent agitation of the water, caused by the 
ascent of a large volume of steam through a small superficies. It 
would be an improvement in boilers, we think, to place over each 
furnace an inverted vessel immerged in the water, which might 
catch the steam in its ascent, and deliver it quietly by a pipe rising 
above the water-level. The water-level would thus be preserved 
from any inconvenient agitation, and the weight of water within the 
boiler would be diminished at the same time that the original depth 
of water over the furnaces was preserved. It would also be an 
improvement to make the sides of the furnaces of marine boilers 
sloping, instead of vertical, as is the common practice, for the steam 
could then ascend freely at the instant of its formation, instead of 
being entangled among the rivets and landings of the plates, and 
superinducing an overheating of the plates by preventing a free 
access of the water to the metal. 

We have, in the following table, collected a few of the principal 
results of experiments made on steam boilers. 



THE STEAM ENGINE. 

Table I. 



215 



NATURE OF THE BOILERS USED. 


ii 


1 
1 

■s 


III 

.§ife 

Sill 


3 

il 


1 

e 

"Si 
il 


■s" 

Ii 
P 


M 

?! 
II 




s 


n 


S 


•< 


^ 


o 


<! 


Cylindrical 










Cylindrical 


Wa^on 1 




with inter- 
nal flue. 


Wagon. 


Wagon. 


Hay -stack. 


tire. 


with inter- 
nal flue. 


with inter- 
nal flue. 


Total area of heated sur- ) 
face in square feet J 


962 


152 


342-8 


459 


334-6 


798 


588 


Length of circuit made by ) 
the heat in feet J 


155 


50-66 


72-5 


52-8 


7-0 


83-1 


78 


Area of fire grates in square > 

feet S 

Weight of fuel burned on] 


23-66 


23-33 


26-09 


35-10 


7-03 


14-25 


37-26 
















each square foot of grate, V 


3-46 


4-00 


10-75 


20-34 


79-33 


46-82 


13-31 


per hour, in lbs. J 
















Cub. ft. of water evaporated] 
















from initial temperature > 


18-87 


16-44 


13-91 


14-11 


11-14 






by 112 lbs. of fuel J 
















Cubic feet of water eva-"| 
















porated per hour from > 


13-81 


13-79 


34-40 


90-7 


55-18 






initial temperature 
















Square feet of heated surO 
















face for each cubic foot of 1 
■water evaporated per j 


69-58 


11-00 


9-96 


5-06 


6-06 


17-17 




hour J 
















Square feet of heated sur-1 
















face for each square foot V 


40-65 
42-2 


6-51 

2-5 


1313 


1308 
1-5 


47-59 
60 


56-0 
15-45 


15-78 


Pressure of steam above ) 
the atmosphere in lbs.- • J 



The economical effects of expansion will be found to be very 
clearly exhibited in the next table. The duties are recorded in the 
fifth line from the top, and the degree of expansion in the bottom 
line. It will be observed, that the order in which the different en- 
gines stand in respect of superiority of duty is the same as in re- 
spect of amount of expansion. The Holmbush engine has a duty 
of 140,484,848 lbs. raised 1 foot by 1 cwt. of coals, and the steam 
acts expansively over '83 of the whole stroke ; while the water- 
works' Cornish engine has only a duty of 105,664,118 lbs., and 
expands the steam over only -687 of the whole stroke. Again, 
comparing the second and last engines together, the Albion Mills 
engine has a duty of 25,756,752 lbs., and no expansive action. 
The water-works' engine, again, acts expansively over one-half of 
its stroke, and has an increased duty of 46,602,333 lbs. Other 
causes, of course, may influence these comparisons, especially the 
last, where one engine is a double-acting rotative engine, and the 
other a single-acting pumping one ; but there can be no doubt that 
the expansive action in the latter is the principal cause of its more 
economical performance. 

The heating surface per horse power allowed by some engineers 
is about 9 square feet in wagon boilers, reckoning the total sur- 
face as effective surface, if the boilers be of a considerable size ; 
but in the case of small boilers, the proportion is larger. The total 



216 



THE PRACTICAL MODEL CALCULATOR. 









! § I S 2 -s 



g '55 S g ^ o 
8-Scllll- 



5-2 



2 c fi^t- 



r-J XO CO CD 

"*0O5 O T-i CO CO 



+ 



CO o 

o o t^ i-i ^ i^ 

I CO r-l 
+ CD,-I 



CO 17I 
CO Th 



o o 

CO '^ 

t^ o co'o" cq 

CM <M I-I 1-1 rH 



+ 



rH CO 

O CS tH 
O 



CO JC^ O 

Tti as 00 00 

00 O tM r-i 



+ 







-*^ 


(>J C5 








000 






•7:$ 


t-T}1 


-* CO CO 





<M CO CO 


CO 




-s 


1^^ 



o o 
o 00 

O CO 



1-1 o 
O CO 
lO rrl O 




THE STEAM ENGINE. 217 

heating surface of a two horse power wagon boiler is, according 
to Fitzgerald's proportions, 30 square feet, or 15 ft. per horse 
power ; whereas, in the case of a 45 horse power boiler the total 
heating surface is 438 square feet, or 9-6 ft. per horse power. 
The capacity of steam room is 8| cubic feet per horse power, in 
the two horse power boiler, and 5f cubic feet in the 20 horse power 
boiler ; and in the larger class of boilers, such as those suitable for 
30 and 45 horse power engines, the capacity of the steam room 
does not fall below this amount, and indeed is nearer 6 than 5f cu- 
bic feet per horse power. The content of water is 18J cubic feet 
per horse power in the two horse power boiler, and 15 cubic feet 
per horse power in the 20 horse power boiler. In marine boilers 
about the same proportions obtain in most particulars. The ori- 
ginal boilers of one or two large steamers were proportioned 
with about half a square foot of fire grate per horse power, and 10 
square feet of flue and furnace surface, reckoning the total amount 
as effective ; but in the boilers of other vessels a somewhat smaller 
proportion of heating surface was adopted. In some cases we 
have found that, in their marine flue boilers, 9 square feet of 
flue and furnace surface are requisite to boil off a cubic foot of 
water per hour, which is the proportion that obtains in some land 
boilers ; but inasmuch as in modern engines the nominal considera- 
bly exceeds the actual power, they allow 11 square feet of heating 
surface per nominal horse power in their marine boilers, and they 
reckon, as effective heating surface, the tops of the flues, and the 
whole of the sides of the flues, but not the bottoms. They have 
been in the habit of allowing for the capacity of the steam space 
in marine boilers 16 times the content of the cylinder ; but as there 
are two cylinders, this is equivalent to 8 times the content of both 
cylinders, which is the proportion commonly followed in land en- 
gines, and which agrees very nearly with the proportion of between 
5 and 6 cubic feet of steam room per horse power. Taking, for 
example, an engine with 23 inches diameter of cylinder and 4 feet 
stroke, which will be 18*4 horse power — the area of the cylinder 
will be 415-476 square inches, which, multiplied by 48, the number 
of inches in the stroke, will give 19942*848 for the capacity of the 
cylinder in cubic inches ; 8 times this is 159542*784 cubic inches, 
or 92-3 cubic feet ; 92*3 divided by 18*4 is rather more than 5 cu- 
bic feet per horse power. There is less necessity, however, that 
the steam space should be large when the flow of steam from the 
boiler is very uniform, as it will be where there are two engines at- 
tached to the boiler at right angles with one another, or where the 
engines work at a great speed, as in the case of locomotive engines. 
A high steam chest too, by rendering boiling over into the steam 
pipes, or priming as it is called, more difficult, obviates the neces- 
sity for so large a steam space ; and the use of steam of a high 
pressure, worked expansively, has the same operation ; so that in 
modern marine boilers, of the tubular construction, where the whole 
of these modifying circumstances exist, there is no necessity, for so 
T 



218 



THE PRACTICAL MODEL CALCULATOR. 



large a proportion of steam room as 5 or 6 cubic feet per horse 
power, and about half that amount more nearly represents the 
general practice. Many allow 0*64 of a square foot per nomi- 
nal horse power of grate bars in their marine boilers, and a good 
effect arises from this proportion ; but sometimes so large an area 
of fire grate cannot be conveniently got, and the proportion of 
half a square foot per horse power seems to answer very well in 
engines working with some expansion, and is now very widely 
adopted. With this allowance, there will be about 22 square feet 
of heating surface per square foot of fire grate ; and if the consump- 
tion of fuel be taken at 6 lbs. per nominal horse power per hour, 
there will be 12 lbs. of coal consumed per hour on each square foot 
of grate. The flues of all flue boilers diminish in their calorimeter 
as they approach the chimney ; some very satisfactory boilers have 
been made by allowing a proportion of 0*6 of a square foot of fire 
grate per nominal horse power, and making the sectional area of 
the flue at the largest part ith of the area of fire grate, and the 
smallest part, where it enters the chimney, j^th of the area of the 
fire grate ; but in some of the boilers proportioned on this plan the 
maximum sectional area is only ^^ or J-, according to the purposes 
of the boiler. These proportions are retained whether the boiler is 
flue or tubular, and from 14 to 16 square feet of tube surface is al- 
lowed per nominal horse power ; but such boilers, although they may 
give abundance of steam, are generally, perhaps needlessly, bulky. 
We shall therefore conclude our remarks upon the subject by 
introducing a table of the comparative evaporative power of differ- 
ent kinds of coal, which will prove useful, by affording data for the 
comparison of experiments upon different boilers when different 
kinds of coal are used. 



Table of the Comparative Evaporative Power of different kinds 

of Coal. 



Description of Coals. 



The best Welsh 

Anthracite American 

The best small Pittsburgh 

Average small Newcastle 

Pennsylvanian 

Coke from Gas-works 

Coke and Newcastle, small, ^ and ^ .... 
Welsh and Newcastle, mixed J and J... 
Derbyshire and small Newcastle, J and J 

Average large NeAvcastle 

Derbyshire 

Blythe Main, Northumberland 



Water evapo- 
rated per lb 
of Coals. 


Lis. 

9-493 


9-14 


8-526 


8-074 


10-45 


7-908 


7-897 


7-865 


7-710 


7-658 


6-772 


6-600 



Strength of boilers. — The extension of the expansive method of 
employing steam to boilers of every denomination, and the gradual 
introduction in connection therewith of a higher pressure than for- 



THE STEAM ENGINE. 



219 



merly, makes the question of the strength of boilers one of great 
and increasing importance. This topic was very successfully eluci- 
dated, a few years ago, by a committee of the Franklin Institute, 
Philadelphia, and we shall here recapitulate a few of the more im- 
portant of the conclusions at which they arrived. Iron boiler plate 
was found to increase in tenacity as its temperature was raised, un- 
til it reached a temperature of 550° above the freezing point, at 
which point its tenacity began to diminish. The following table 
exhibits the cohesive strength at different temperatures. 



At 32° to 80° the tenacity was 

At 570° 

At 720° 

At 1050° 

At 1240° 

At 1317° 

At 3000° iron becomes fluid. 



= 56,000 lbs., or l-7t]a below" its maximum. 
= 66,500 lbs., the maximum. 
= 55,000 lbs., the same nearly as at 32°. 
= 32,000 lbs., nearly |- of the maximum. 
= 22,000 lbs., nearly ^ of the maximum. 
= 9,000 lbs., nearly l-7th of the maximum. 



The difference in strength between strips of iron cut in the di- 
rection of the fibre, and strips cut across the grain, was found to 
be about 6 per cent, in favour of the former. Repeated piling and 
welding was found to increase the tenacity and closeness of the 
iron, but welding together different kinds of iron was found to give 
an unfavourable result ; riveting plates was found to occasion a 
diminution in their strength, to the extent of about one-third. The 
accidental overheating of a boiler was found to reduce its strength 
from 65,000 lbs. to 45,000 lbs. per square inch. Taking into ac- 
count all these contingencies, it appears expedient to limit the ten- 
sile force upon boilers in actual use to about 3000 lbs. per square 
inch of iron. 

Copper follows a different law, and appears to diminish in strength 
by every addition of heat, reckoning from the freezing point. The 
square of the diminution of strength seems to keep pace with the 
cube of the temperature, as appears by the following table : — 

Table showing the Diminution of Strength of Copper Boiler 
Plates hy additions to the Temperature, the Cohesion at 32° being 
32,800 lbs. per Square Inch. 



No. 


Temperature 


Diminution of 


No 


Temperature 


Diminution of 


above 320. 


Strength. 




above 320. 


Strength. 


1 


90° 


0-0175 


9 


660° 


0-3425 


2 


180 


0540 


10 


769 


0-4398 


3 


270 


0-0926 


11 


812 


0-4944 


4 


360 


0-1513 


12 


880 


0-5581 


5 


450 


0-2046 


13 


984 


0-6691 


6 


460 


0-2133 


14 


1000 


0-6741 


7 


513 


0-2446 


15 


1200 


0-8861 


8 


529 


0-2558 


16 


1300 


1-0000 



In the case of iron, the following are the results when tabulated 
after a similar fashion. 



220 



THE PRACTICAL MODEL CALCULATOR. 



Table of Experiments on Iron Boiler Plate at High Tempera- 
ture ; the Mean Maximum Tenacity being at 550° = 65,000 lbs. 
per Square Inch. 



Temperature 


Diminution of 


Temperature 


Diminution of 


observed. 


Tenacity observed. 


observed. 


Tenacity observed. 


650° 


0-0000 


824° 


0-2010 


570 


0-0869 


932 


0-3324 


596 


0-0899 


947 


0-3593 


600 


0-0964 


1030 


0-4478 


630 


0-1047 


1111 


0-5514 


562 


0-1155 


1155 


0-6000 


722 


0-1436 


1159 


0-6011 


732 


0-1491 


1187 


0-6352 


734 


0-1535 


1237 


0-6622 


766 


0-1589 


1245 


0-6715 


770 


0-1627 


1317 


0-7001 



The application of stays to marine boilers, especially in those 
parts of the water spaces which lie in the wake of the furnace bars, 
has given engineers much trouble ; the f plate, of which ordinary 
boilers are composed, is hardly thick enough to retain a stay with 
security by merely tapping the plate, whereas, if the stay be ri- 
veted, the head of the rivet will in all probability be soon burnt 
away. The best practice appears to be to run the stays used for 
the water spaces in this situation, in a line somewhat beneath the 
level of the bars, so that they may be shielded as much as possible 
from the fire, while those which are required above the level of the 
bars should be kept as nearly as possible towards the crown of the 
furnace, so as to be removed from the immediate contact of the fire. 
Screw bolts with a fine thread tapped into the plate, and with a 
thin head upon the one side, and a thin nut made of a piece of 
boiler plate on the other, appear to be the best description of stay 
that has yet been contrived. The stays between the sides of the 
boiler shell, or the bottom of the boiler and the top, present little 
difficulty in their application, and the chief thing that is to be at- 
tended to is to take care that there be plenty of them ; but we may 
here remark that we think it an indispensable thing, when there is 
any high pressure of steam to be employed, that the furnace crown 
be stayed to the top of the boiler. This, it will be observed, is done 
in the boilers of the Tagus and Infernal ; and we know of no better 
specimen of staying than is afi"orded by those boilers. 

area of steam passages. 

Rule. — To the temperature of steam in the boiler add the con- 
stant increment 459 ; multiply the sum by 11025 ; and extract the 
square root of the product. Multiply the length of stroke by the 
number of strokes per minute ; divide the product by the square 
root just found ; and multiply the square root of the quotient by 
the diameter of the cylinder ; the product will be the diameter of 
the steam passages. 



THE STEAM ENGINE. 221 

Let it be required to determine the diameter of the steam pas- 
sages in an engine of which the diameter of the cylinder is 48 
inches, the length of stroke 4|- feet, and the number of strokes per 
minute 26, supposing the temperature under which the steam is 
generated to be 250 degrees of Pahrenheit's thermometer. 

Here by the rule we get ^^11025(250 -f 459) = 2795-84 ; the 
number of strokes is 26, and the length of stroke 4 J feet ; hence 

/ 117 
it is 5 = ^V 2795'84 "" ^"20456^ = 0-20456 X 48 = 9-819 inches ; 

so that the diameter of the steam passages is a little more than one- 
fifth of the diameter of the cylinder. The same rule will answer 
for high and low pressure engines, and also for the passages into 
the condenser. 

LOSS OF FORCE BY THE DECREASE OF TEMPERATURE IN THE STEAM PIPES. 

Rule. — From the temperature of the surface of the steam pipes 
subtract the temperature of the external air ; multiply the remain- 
der by the length of the pipes in feet, and again by the constant 
number or coefficient 1-68 ; then divide the product by the diameter 
of the pipe in inches drawn into the velocity of the steam in feet 
per second, and the quotient will express the diminution of tem- 
perature in degrees of Fahrenheit's thermometer. 

Let the length of the steam pipe be 16 feet and its diameter 5 
inches, and suppose the velocity of the steam to be about 95 feet 
per second, what will be the diminution of temperature, on the sup- 
position that the steam is at 250° and the external air at 60° of 
Fahrenheit ? 

Here, by the note to the above rule, the temperature of the sur- 
face of the steam pipe is 250 — 250 X 0*05 = 237*5 ; hence we get 
^„ ^ 1-68 X 16(237-5 -60) ^ j^.^^^ ^^^^^^_ 

O X oO 

If we examine the manner of the composition of the above equa- 
tion, it will be perceived that, since the diameter of the pipe and 
the velocity of motion enter as divisors, the loss of heat will be less 
as these factors are greater ; but, on the other hand, the loss of 
heat will be greater in proportion to the length of pipe and the 
temperature of the steam. Since the steam is reduced from a 
higher to a lower temperature during its passage through the steam 
pipes, it must be attended with a corresponding diminution in the 
elastic force ; it therefore becomes necessary to ascertain to what 
extent the force is reduced, in consequence of the loss of heat that 
takes place in passing along the pipes. This is an inquiry of some 
importance to the manufacturers of steam engines, as it serves to 
guard them against a very common mistake into which they are 
liable to fall, especially in reference to steamboat engines, where it 
is usual to cause the pipe to pass round the cylinder, instead of 
carrying it in the shortest direction from the boiler, in order to de- 
crease the quantity of surface exposed to the cooling effect of the 
atmosphere. 
t2 



222 THE PRACTICAL MODEL CALCULATOR. 

Rule. — From the temperature of the surface of the steam pipe 
subtract the temperature of the external air ; multiply the remain- 
der by the length of the pipe in feet, and again by the constant 
fractional coefficient 0*00168 ; divide the product by the diameter 
of the pipe in inches drawn into the velocity of steam in feet per 
second, and subtract the quotient from unity ; then multiply the 
difference thus obtained by the elastic force corresponding to the 
temperature of steam in the boiler, and the product will be the 
elastic force of the steam as reduced by cooling in passing through 
the pipes. 

Let the dimensions of the pipe, the temperature of the steam, 
and its velocity through the passages, be the same as in the pre- 
ceding example, what will be the quantity of reduction in the elastic 
force occasioned by the effect of cooling in traversing the steam 
pipe? 

Since the elastic force of the steam in the boiler enters the equa- 
tion from which the above rule is deduced, it becomes necessary in 
the first place to calculate its value ; and this is to be done by a 
rule already given, which answers to the case in which the tempera- 
ture is greater than 212° ; thus we have 

250 X 1-69856 = 424-640 
Constant number = 205*526 add 

Sum = 630-166 log. 2-79945 

Constant divisor = 333 log. 2-522444 subtract 

0-277011 X 6-42 = 1-778410, 
which is the logarithm of 60*036 inches of mercury. 

Again, we have 250 — 0-05 X 250 = 237*5 ; consequently, by 
multiplying as directed in the rule, we get 237-5 x 0-00168 x 16 
= 6-384, which being divided by 95 x 5 = 475, gives 0-01344 ; and 
by taking this from unity and multiplying the remainder by the 
elastic force as calculated above, the value of the reduced elastic 
force becomes 

/ = 60-036 (1 - 0-01344) = 59-229 inches of mercury. 

The loss of force is therefore 60-036 - 59-229 = 0-807 inches of 
mercury, which amounts to ^th part of the entire elastic force of 
the steam in the boiler as generated under the given temperature, 
being a quantity of sufficient importance to claim the attention of 
our engineers. 

FEED WATER. 

The quantity of water required to supply the waste occasioned 
by evaporation from a boiler, or, as it is technically termed, the 
" feed water" required by a boiler working with any given pressure, 
is easily determinable. For, since the relative volumes of water 
and steam at any given pressure are known, it becomes necessary 
merely to restore the quantity of water by the feed pump equiva- 



THE STEAM ENGINE. 223 

lent to that abstracted in tlie form of steam, which the known rela- 
tion of the density to the pressure of the steam renders of easy 
accomplishment. In practice, however, it is necessary that the 
feed pump should be able to supply a much larger quantity of water 
than what theory prescribes, as a great waste of water sometimes 
occurs from leakage or priming, and it is necessary to provide 
against such contingencies. The feed pump is usually made of 
such dimensions as to be capable of supplying 3J times the water 
that the boiler will evaporate, and in low pressure engines, where 
the cylinder is double acting and the feed pump single acting, this 
proportion will be maintained by making the pump a 240th of the 
capacity of the cylinder. In low pressure engines the pressure in 
the boiler may be taken at 5 lbs. above the pressure of the atmo- 
sphere, or 20 lbs. in all ; and as high pressure steam is merely low 
pressure steam compressed into a smaller compass, the size of the 
feed pump relatively to the size of the cylinder must obviously vary 
in the direct proportion of the pressure. If, then, the feed pump 
be l-240th of the capacity of the cylinder when the total pressure 
of the steam is 20 lbs., it must be l-120th of the capacity of the 
cylinder when the total pressure of the steam is 40 lbs., or 25 lbs. 
above the atmosphere. This law of variation is expressed by the fol- 
lowing rule, which gives the capacity of feed pump proper for all 
pressures : — Multiply the capacity of the cylinder in cubic inches by 
the total pressure of the steam in lbs. per square inch, or the pressure 
in lbs. per square inch on the safety valve, plus 15, and divide the 
product by 4800 ; the quotient is the capacity of the feed pump in 
cubic inches, when the feed pump is single acting and the engine 
double acting. If the feed pump be double acting, or the engine 
single acting, the capacity of the pump must be just one-half what 
is given by this rule. 

CONDENSING WATER. 

It was found that the most beneficial temperature of the hot 
well was 100 degrees. If, therefore, the temperature of the 
steam be 212°, and the latent heat 1000°, then 1212° may be 
taken to represent the heat contained in the steam, or 1112° 
if we deduct the temperature of the hot well. If the tempera- 
ture of the injection water be 50°, then 50 degrees of cold are 
available for the abstraction of heat, and as the total quantity of 
heat to be abstracted is that requisite to raise the quantity of water 
in the steam 1112 degrees, or 1112 times that quantity, one degree, 
it would raise one-fiftieth of this, or 22*24 times the quantity of 
water in the steam, 50 degrees. A cubic inch of water, therefore, 
raised into steam, will require 22*24 cubic inches of water at 50 
degrees for its condensation, and will form therewith 23*24 cubic 
inches of hot water at 100 degrees. It has been a practice to 
allow about a wine pint (28*9 cubic inches) of injection water for 
every cubic inch of water evaporated from the boiler. The usual 
capacity for the cold water pump is Jgth of the capacity of the 
cylinder, which allows some water to run to waste. As a maximum 



224 THE PRACJTICAL MODEL CALCULATOR. 

effect is obtained wlien the temperature of tlie hot well is about 
100°, it will not be advisable to reduce it below that temperature 
in practice. With the superior vacuum due to a temperature of 70° 
or 80° the admission of so much cold water into the condenser 
becomes necessary, — and which has afterwards to be pumped out 
in opposition to the pressure of the atmosphere, — so that the gain in 
the vacuum does not equal the loss of power occasioned by the 
additional load upon the pump, and there is, therefore, a clear loss 
by the reduction of the temperature below 100°, if such reduction 
be caused by the admission of an additional quantity of water. If 
the reduction of temperature, however, be caused by the use of 
colder water, there is a gain produced by it, though the gain will 
within certain limits be greater, if advantage be taken of the low- 
ness of the temperature to diminish the quantity of injection. 



SAFETY VALVES. 

Rule. — Add 459 to the temperature of the steam in degrees 
of Fahrenheit ; divide the sum by the product of the elastic force 
of the steam in inches of mercury, into its excess above the weight 
of the atmosphere in inches of mercury ; multiply the square root 
of the quotient by '0653 ; multiply this product by the number of 
cubic feet per hour of water evaporated, and this last product is 
the theoretical area of the orifice of the safety valve in square 
inches. 

To apply this to an example — which, however, it must be remem- 
bered, will give a result much too small for practice. 

Required the least area of a safety valve of a boiler suited for a 
250 horse power engine, working with steam 6 lbs. more than the 
atmosphere on the square inch. 

In this case the total pressure is equal to 21 lbs. per square 
inch ; and as in round numbers one pound of pressure is equal to 
about two inches of mercury, it follows that / = 42 inches of 
mercury. 

It will be necessary to calculate t from formula (S) already given. 
The operation is as follows : — 

log. 42 -^ 6-42 = 1-623249 -^ 6-42 = 0-252842 
constant co-efficient = 196 2-292363 

2-545205 
natural number = 350-92 
constant temperature = 121 

t = 229-92 
^ ^ /459+^ / 459 + 229-92 
therefore ^ ytj^tW) = J 42 x 12 



-J 



/(/-30) 
688-92 



^^^ = ^ 1-3669 = 1-168; 
therefore x = -0653 x 1-168 x N = -0757 N. 



THE 6TEAM ENGINE. 2£5 

"VYe have stated in a former part of this Tiork that a cubic foot 
of water evaporated per hour is equivalent to one horse power; 
therefore in this case N = 250 and x = 18*925 sq. in. 

As another example. Required the proper area of the safety 
valve of a boiler suited to an engine of 500 horse power, when it 
is wished that the steam should never acquire an elastic force 
greater than 60 lbs. on the square inch above the atmosphere. 

In this case the whole elastic force of the steam is 75 lbs. ; and 
as 1 pound corresponds in round numbers to 2 inches of mercury, 
it follows that / = 150. It will be necessary to calculate the 
temperature corresponding to this force. The operation is as 
follows : — 

Log. 150 ^ 6-42 = 2-176091 -r- 642 = -338955 
constant co-efficient = 196 log. 2-292363 add 

natural number = 427*876 2-631318 

constant temperature = 121 

required temperature 306*876 degrees of Fahrenheit's scale 

459 + f 459 + 306*876 765*876 765-896 
therefore ^ ^^ __ g^^ - ^^^ ^^^ _ ^^-^ - 15Q ^ 120 " 18000 

/ 459 -f ^ 

= -043549 ; therefore J f ,f _ ^^\ = V -042549 = -20628. 

Hence the required area = -0653 X -20628 X 500 = -01347 X 
500 = 6-735 square inches. 

If the area of the safety valve of a boiler suited for an engine 
of 500 horse power be required, when it is wished the steam should 
never acquire a greater temperature than 300°, it will be necessary 
to calculate the elastic force corresponding to this temperature ; and 
by formula for this purpose, the required area = -0653 x -231 X 
500 = -0151 X 500 = 7*55 square inches. It will be perceived 
from these examples that the greater the elasticity and the higher 
the corresponding temperature the less is the area of the safety 
valve. This is just as might have been expected, for then the 
steam can escape with increased velocity. We may repeat that the 
results we have arrived at are much less than those used in practice. 
For the sake of safety, the orifices of the safety valve are inten- 
tionally made much larger than what theory requires ; usually {-^ of 
a square inch per horse power is the ordinary proportion allowed 
in the case of low pressure engines. 

THE SLIDE VALVE. 

The four following practical rules are applicable alike to short 
slide and long D valves. 

Rule I. — To find how much cover must he given on the steam 
side in order to cut the steam off at any given part of the stroke. — 
From the length of the stroke of the piston, subtract the length of 
that part of the stroke that is to be made before the steam is cut 
oflf. Divide the remainder by the length of the stroke of the 

15 



226 THE PRACTICAL MODEL CALCULATOR. 

piston, and extract the square root of the quotient. Multiply the 
square root thus found by half the length of the stroke of the valve, 
and from the product take half the lead, and the remainder will be 
the cover required. 

Rule II. — To find at what part of the stroke any given amount 
of cover on the steam side will cut off the steam. — Add the cover 
on the steam side to the lead ; divide the sum by half the length 
)f stroke of the valve. In a table of natural sines find the arc 
n^hose sine is equal to the quotient thus obtained. To this arc add 
90°, and from the sum of these two arcs subtract the arc whose 
cosine is equal to the covjr on the steam side divided by half the 
stroke of the valve. Find the cosine of the remaining arc, add 1 
to it, and multiply the sum by half the stroke of the piston, and 
the product is the length of that part of the stroke that will be 
made by the piston before the steam is cut off. 

HuLE III. — To find how much before the end of the stroke, the 
exhaustion of the steam in front of the piston will be cut off. — To 
the cover on the steam side add the lead, and divide the sum by 
half the length of the stroke of the valve. Find the arc whose sine 
is equal to the quotient, and add 90° to it. Divide the cover on 
the exhausting side by half the stroke of the valve, and find the arc 
whose cosine is equal to the quotient. Subtract this arc from the 
one last obtained, and find the cosine of the remainder. Subtract 
this cosine from 2, and multiply the remainder by half the stroke 
of the piston. The product is the distance of the piston from the 
end of its stroke when the exhaustion is cut off. 

Rule IV. — To find how far the piston is from the end of its 
stroke, when the steam that is propelling it by expansion is allowed 
to escape to the condenser. — To the cover on the steam side add the 
lead, divide the sum by half the stroke of the valve, and find the 
arc whose sine is equal to the quotient. Find the arc whose cosine 
is equal to the cover on the exhausting side, divided by half the 
stroke of the valve. Add these two arcs together, and subtract 
90°. Find the cosine of the residue, subtract it from 1, and mul- 
tiply the remainder by half the stroke of the piston. The product 
is the distance of the piston from the end of its stroke, when the 
steam that is propelling it is allowed to escape to the condenser. 
In using these rules, all the dimensions are to be taken in inches, 
and the answers will be found in inches also. 

From an examination of the formulas we have given on this 
subject, it will be perceived (supposing that there is no lead) that 
the part of the stroke where the steam is cut off, is determined by 
the proportion which the cover on the steam side bears to the 
length of the stroke of the valve : so that in all cases where the 
cover bears the same proportion to the length of the stroke of the 
valve, the steam will be cut off at the same part of the stroke of 
the piston. 

In the first line, accordingly, of Table I., will be found eight 
iifferent parts of the stroke of the piston designated ; and directly 



THE STEAM ENGINE. 



227 



below each, in the second line, is given the quantity of cover requi- 
site to cause the steam to be cut off at that particular part of the 
stroke. The different sizes of the cover are given in the second 
line, in decimal parts of the length of the stroke of the valve ; so 
that, to get the quantity of cover corresponding to any of the given 
degrees of expansion, it is only necessary to take the decimal in 
the second line, which stands under the fraction in the first, that 
marks the degree of expansion, and multiply that decimal by the 
length you intend to make the stroke of the valve. Thus, suppose 
you have an engine in which you wish to have the steam cut off 
when the piston is a quarter of the length of its stroke from the 
end of it, look in the table, and you will find in the third column 
from the left, ^. Directly under that, in the second line, you have 
the decimal -250. Suppose that you think 18 inches will be a con- 
venient length for the stroke of the valve, multiply the decimal 
•250 by 18, which gives 4|. Hence we learn that with an 18 inch 
stroke for the valve, 4J inches of cover on the steam side will cause 
the steam to be cut off when the piston has still a quarter of its 
stroke to perform. 

Half the stroke of the valve must always be at least equal to the 
cover on the steam side added to the breadth of the port. By the 
"breadth" of the port, we mean its dimension in the direction of 
the valve's motion ; in short, its perpendicular depth when the 
cylinder is upright. The words "cover" and "lap" are synony- 
mous. Consequently, as the cover, in this case, must be 4| inches, 
and as half the stroke of the valve is 9 inches, the breadth of the 
port cannot be more than (9 — 4^ = 4^) 4J inches. If this 
breadth of port is not enough, we must increase the stroke of the 
valve ; by which means we shall get both the cover and the breadth 
of the port proportionally increased. Thus, if we make the length 
of valve stroke 20 inches, we shall have for the cover -250 X 20 = 5 
inches, and for the breadth of the port 10 — 5 = 5 inches. 

Table I. 



Distance of the piston from "] 
the termination of its 
stroke, when the steam V 
is cut off, in parts of the 
length of its stroke. J 


or 


7 


or 


^ 


or 


or 


or 


•102 


Cover on the steam side of ■] 
the valve, in decimal 1 
parts of the length of its 
stroke. 


•289 


•270 


•250 


•228 


•204 


•177 


■144 



This table, as we have already intimated, is computed on the 
supposition that the valve is to have no lead ; but, if it is to have 
lead, all that is necessary is to subtract half the proposed lead from 
the cover found from the table, and the remainder will be the 



228 



THE PRACTICAL MODEL CALCULATOR. 



proper quantity of cover to give to the valve. Suppose that, in 
the last example, the valve was to have ^ inch of lead, we would 
subtract | inch from the 5 inches found for the cover by the table : 
that would leave 4| inches for the quantity of cover that the valve 
ought to have. 

Table II. 



Length of 
the stroke 


Cover required 


on the steam side of the valve to cut the steam off at any of the | 






under-noted parts of the stroke. 






of the valve. 
Inches. 


















I 


A 


I 


A 


I 


1 


' 1^2 


^ 


24 


6-94 


6-48 


6^00 


5-47 


4-90 


4-25 


3-47 


2^45 


23A 


6-79 


6-34 


5^88 


5-36 


4-79 


4^16 


3^39 


2^39 


23 


6-65 


6-21 


5-75 


6-24 


4-69 


4-07 


3^32 


2-34 


22A 


6-50 


6-07 


6-62 


5^13 


4-59 


3-98 


3-25 


2^29 


22 


6-36 


5-94 


5-50 


5-02 


4-49 


3^89 


3-13 


2^24 


21J 


6-21 


5-80 


6^38 


4-90 


4-39 


3-80 


3-10 


2^19 


' 21 


6-07 


5-67 


5-25 


4-79 


4-28 


3^72 


3-03 


2^14 


20i- 


5-92 


5-53 


5-12 


4-67 


4^18- 


3-63 


2-96 


2^09 


.' 20" 


5-78 


5-40 


5^00 


4^56 


4^08 


3-54 


2-89 


2-04 


V 19J 


5-64 


5-26 


4-87 


4-45 


3-98 


3-45 


2-82 


1-99 


r 19 


5-49 


5-13 


4-75 


4^33 


3-88 


3-36 


2^74 


1^94 


i 1% 


5-34 


4-99 


4-62 


4-22 


3-77 


3-27 


2-67 


1-88 


18 


5-20 


4-86 


4-50 


4-10 


3-67 


3-19 


2^60 


1^83 


17J 


5-06 


4-72 


4-37 


3-99 


3^57 


3-10 


2^53 


1-78 


17 


4-91 


4-.59 


4-25 


3^88 


3^47 


3^01 


2-45 


173 


16J 


4-77 


4-45 


4-12 


3^76 


3-36 


2-92 


2-38 


1-68 


16 


4-62 


4-32 


4-00 


3-65 


3-26 


2^83 


2-31 


1^63 


15i 


4-48 


4-18 


3-87 


3-53 


3-16 


2-74 


2-24 


1^58 


15 


4-33 


4-05 


3-75 


3-42 


3^06 


2-65 


2-16 


1^53 


14^ 


4-19 


3-91 


3-62 


3-31 


2^96 


2-57 


2-09 


1^48 


14 


4-05 


3-78 


3-50 


3-19 


2-86 


2-48 


2-02 


1-43 


13^ 


3-90 


3-64 


3-37 


3^08 


2-75 


2-39 


1-95 


1-37 


13 


3-76 


3-51 


3-25 


2-96 


2^65 


2-30 


1^88 


1-32 


12J 


3-61 


3-37 


3-12 


2-85 


2^55 


2^21 


1-80 


1-27 


12 


3-47 


3-24 


3-00 


2-74 


2^45 


2-12 


1-73 


1-22 


11* 


3-32 


3-10 


2-87 


2-62 


2^35 


2-03 


1-66 


1^17 


11 


3-18 


2-97 


2-75 


2-51 


2-24 


1-95 


1^58 


1^12 


10* 


3-03 


2-83 


2-62 


2^39 


2-14 


1-86 


1^51 


1-07 


10 


2-89 


2-70 


2-50 


2-28 


2^04 


1^77 


1-44 


102 


9* 


2-65 


2-56 


2-37 


2-17 


1^93 


1^68 


1-32 


•96 


9 


2-60 


2-43 


2-25 


2^05 


1-84 


1^59 


1^30 


•92 


H 


2-46 


2-29 


2^12 


1-94 


1^73 


1^50 


1-23 


•86 


8 


2-31 


2-16 


2-00 


1-82 


1-63 


1^42 


1^15 


•81 


n 


2-16 


2-02 


1-87 


1^71 


1^53 


1^33 


1-08 


•76 


7 


2-02 


1-89 


1-75 


1^60 


1^43 


1^24 


l^Ol 


•71 


^ 


1-88 


1-75 


1-62 


1-48 


1-32 


1^15 


•94 


•66 


6 


1-73 


1-62 


1-50 


1-37 


1^22 


1^06 


•86 


•61 


5* 


1-58 


1-48 


1-37 


1-25 


1-12 


•97 


•79 


•56 


5 


1-44 


1-35 


1^25 


1^14 


1-02 


•88 


•72 


•51 


H 


1-30 


1-21 


1-12 


1^03 


•92 


•80 


•65 


•46 


4 


1-16 


1-08 


1-00 


•91 


•82 


•71 


•58 


•41 


3* 


1-01 


•94 


•87 


•80 


•71 


•62 


•50 


•35 


3 


•86 


•81 


•75 


•68 


•61 


•53 


•44 


•30 



Table II. is an extension of Table I. for the purpose of obviating, 
in most cases, the necessity of even the very small degree of 
trouble required in multiplying the stroke of the valve by one of 
the decimals in Table I. The first line of Table II. consists, as in 
Table I., of eight fractions, indicating the various parts of the stroke 



THE STEAM ENGINE. 229 

at which the steam may be cut off. The first column on the left 
hand consists of various numbers that represent the different 
lengths that may be given to the stroke of the valve, diminishing, 
by half-inches, from 24 inches to 3 inches. Suppose that you wish 
the steam cut off at any of the eight parts of the stroke indicated 
in the first line of the table, (say at | from the end of the stroke,) 
you find | at the top of the sixth column from the left. Look for 
the proposed length of stroke of the valve (say 17 inches) in the 
first column on the left. From 17, in that column, run along the 
line towards the right, and in the sixth column, and directly under 
the I at the top, you will find 3*47, which is the cover required to 
cause the steam to be cut off at | from the end of the stroke, if the 
valve has no lead. If you wish to give it lead, (say ^ inch,) sub- 
tract the half of that, or | = -125 inch from 3*47, and you will have 
3*47 — "125 = 3*345 inches, the quantity of cover that the valve 
should have. 

To find the greatest breadth that we can give to the port in this 
case, we have, as before, half the length of stroke, 8J— 3'345=5-155 
inches, which is the greatest breadth we can give to the port with 
this length of stroke. It is scarcely necessary to observe that it is 
not at all essential that the port should be so broad as this ; indeed, 
where great length of stroke in the valve is not inconvenient, it is 
always an advantage to make it travel farther than is just neces- 
sary to make the port full open; because, when it travels farther, 
both the exhausting and steam ports are more quickly opened, so 
as to allow greater freedom of motion to the steam. 

The manner of using this table is so simple, that we need not 
trouble the reader with more examples. We pass on, therefore, to 
explain the use of Table III. 

Suppose that the piston of a steam engine is making its down- 
ward stroke, that the steam is entering the upper part of the cylin- 
der by the upper steam-port, and escaping from below the piston 
by the lower exhausting-port; then, if (as is generally the case) 
the slide valve has some cover on the steam side, the upper port 
will be closed before the piston gets to the bottom of the stroke, 
and the steam above then acts expansively, while the communica- 
tion between the bottom of the cylinder and the condenser still 
continues open, to allow any vapour from the condensed water in 
the cylinder, or any leakage past the piston, to escape into the 
condenser ; but, before the piston gets to the bottom of the cylin- 
der, this passage to the condenser will also be cut off by the valve 
closing the lower port. Soon after the lower port is thus closed, 
the upper port will be opened towards the condenser, so as to allow 
the steam that has been acting expansively to escape. Thus, be- 
fore the piston has completed its stroke, the propelling power is 
removed from behind it, and a resisting power is opposed before it, 
arising from the vapour in the cylinder, which has no longer any 
passage open to the condenser. It is evident, that if there is no 
cover on the exhausting side of the valve, the exhausting port before 



k 



230 THE PRACTICAL MODEL CALCULATOR. 

the piston will be closed, and the one behind it opened, at the same 
time ; but, if there is any cover on the exhausting side, the port 
before the piston will be closed before that behind it is opened ; and 
the interval between the closing of the one and the opening of 
the other will depend on the quantity of cover on the exhausting 
side of the valve. Again, the position of the piston in the cylin- 
der, when these ports are closed and opened respectively, will 
depend on the quantity of cover that the valve has on the steam 
side. If the cover is large enough to cut the steam off when the 
piston is yet a considerable distance from the end of its stroke, 
these ports will be closed and opened at a proportionably early part 
of the stroke ; and when it is attempted to obtain great expansion 
by the slide-valve alone, without an expansion-valve, considerable 
loss of power is incurred from this cause. 

Table III. is intended to show the parts of the stroke where, un- 
der any given arrangement of slide valve, these ports close and open 
respectively, so that thereby the engineer may be able to estimate 
how much of the efficiency of the engine he loses, while he is trying 
to add to the power of the steam by increasing the expansion in 
this manner. In the table, there are eight double columns, and at 
the heads of these columns are eight fractions, as before, represent- 
ing so many different parts of the stroke at which the steam may 
be supposed to be cut off. 

In the left-hand single column in each double one, are four deci- 
mals, which represent the distance of the piston (in terms of the 
length of its stroke) from the end of its stroke when the exhausting- 
port before it is opened, corresponding with the degree of expansion 
indicated by the fraction at the top of the double column and the 
cover on the exhausting side opposite to these decimals respectively 
in the left-hand column. The right-hand single column in each 
double one contains also each four decimals, which show in the same 
way at what part of the stroke the exhausting-port behind the pis- 
ton is opened. A few examples will, perhaps, explain this best. 

Suppose we have an engine in which the slide valve is made to 
cut the steam off when the piston is l-3d from the end of its stroke, 
and that the cover on the exhausting side of the valve is l-8th of 
the whole length of its stroke. Let the stroke of the piston be 6 
feet, or 72 inches. We wish to know when the exhausting-port 
before the piston will be closed, and when the one behind it will be 
opened. At the top of the left-hand double column, the given de- 
gree of expansion (l-3d) is marked, and in the extreme left column 
we have at the top the given amount of cover (l-8th). Opposite the 
l-8th, in the first double column, we have '178 and -033, w^hich 
decimals, multiplied respectively by 72, the length of the stroke, 
will give the required positions of the piston: thus 72 X •178=12-8 
inches = distance of the piston from the end of the stroke when the 
exhausting-port hefore the piston is shut ; and 72 X -033 = 2*38 
inches = distance of the piston from the end of its stroke when the 
exhausting-port behind it is opened. 



THE STEAM ENGINE. 



231 



l-8th 

l-16th 

l-32d 




Cover on the exhausting side of the valve in parts of 
the length of its stroke. 


O i-i 1— ' h-i 
CD I-' CO "M 
to CO O 00 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 




o o o o 

CO ^ OS CO 

to CO O CO 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


6 ^^ ^^ ^ 

00 O H-' OS 
to 1— OO •-' 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 




•026 
•052 
•066 
•082 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


OS 00 o *- 

•^ en o CO 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 


TO » Hi. B 

'III 


§ ^ ^ 2 

^ >-i O CD 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
ft is opened (in parts of the stroke). 


O O O (^ 

§^ s ^ g 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 




g g g 2 

en to o to 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


ii^.% 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 


^§ SB 


rf». CO to o 
CO CO to oo 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


o o o o 

CO Hi. en CD 

CO CO 00 CO 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 


f ? 

fl> f-l p 

sS.?g 

&B^ 


g g 2 8 

CO CO en Hi. 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


•074 
•043 
;033 
•022 


Distance of the piston from the enu of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke.) 


•;b§^ 

CD P 


g 2 § § 

to CO 00 t-J 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 


2 8 g g 

l-« hf». ^I CO 


Distance of the piston from the end of its 
stroke, when the exhausting-port before 
it is shut (in parts of the stroke). 




2 § 8 8 

h-i *. to I-' 


Distance of the piston from the end of its 
stroke, when the exhausting-port behind 
it is opened (in parts of the stroke). 






232 THE PRACTICAL MODEL CALCULATOR. 

To take another example. Let the stroke of the valve be 16 
inches, the cover on the exhausting side J inch, the cover on the 
steam side 3^ inches, the length of the stroke of the piston 60 inches. 
It is required to ascertain all the particulars of the working of this 
valve. The cover on the exhausting side is evidently ^ of the 
length of the valve stroke. Again, looking at 16 in the left-hand 
column of Table II., we find in the same horizontal line 3*26, or very 
nearly 3^ under ^ at the head of the column, thus showing that the 
steam will be cut ofi" at ^ from the end of the stroke. Again, under 
i at the head of the fifth double column from the left in Table III., 
and in a horizontal line with ^ in the left-hand column, we have 
•053 and -033. Hence, -053 X 60 = 3-18 inches = distance of the 
piston from the end of its stroke when the exhausting-port before 
it is shut, and '033 x 60 = 1*98 inches = distance of the piston 
from the end of its stroke when the exhausting-port behind it is 
opened. If in this valve the cover on the exhausting side were 
increased (say to 2 inches, or i of the stroke,) the effect would be to 
make the port before the valve be shut sooner in the proportion of 
•109 to ^053, and the port behind it later in the proportion of '008 
to ^033 (see Table III.) Whereas, if the cover on the exhausting 
side were removed entirely, the port before the piston would be 
shut and that behind it opened at the same time, and (see bottom 
of fifth double column. Table III.) the distance of the piston from 
the end of its stroke at that time would be -043 x 60 = 2*58 inches. 

An inspection of Table III. shows us the effect of increasing the 
expansion by the slide-valve in augmenting the loss of power occa- 
sioned by the imperfect action of the eduction passages. Referring 
to the bottom line of the table, we see that the eduction passage 
before the piston is closed, and that behind it opened, (thus destroy- 
ing the whole moving power of the engine,) when the piston is '092 
from the end of its stroke, the steam being cut off at J from the 
end. Whereas, if the steam is only cut off at ^ from the end of 
the stroke, the moving power is not withdrawn till only 'Oil of the 
stroke remains uncompleted. It will also be observed that in- 
creasing the cover on the exhausting side has the effect of retaining 
the action of the steam longer heliind the piston, but it at the same 
time causes the eduction-port before it to be closed sooner. 

A very cursory examination of the action of the slide valve is 
sufficient to show that the cover on the steam side should always be 
greater than on the exhausting side. If they are equal, the steam 
would be admitted on one side of the piston at the same time that 
it was allowed to escape from the other ; but universal experience 
has shown that when this is the case, a very considerable part of 
the power of the engine is destroyed by the resistance opposed to 
the piston, by the exhausting steam not getting away to the con- 
denser with sufficient rapidity. Hence we see the necessity of 
the cover on the exhausting side being always less than the cover on 
the steam side ; and the difference should be the greater the higher 
the velocity of the piston is intended to be, because the quicker the 



THE STEAM ENGINE. 



233 



piston moves the passage for the waste steam requires to be the 
larger, so as to admit of its getting away to the condenser with as 
great rapidity as possible. In locomotive or other engines, where 
it is not wished to expand the' steam in the cylinder at all, the slide 
valve is sometimes made with very little cover on the steam side : 
and in these circumstances, in order to get a sufficient diiference 
between the cover on the steam and exhausting sides of the valve, 
it may be necessary not only to take away all the cover on the 
exhausting side, but to take off still more, so as to make both ex- 
hausting passages be in some degree open, when the valve is at the 
middle of its stroke. This, accordingly, is sometimes done in such 
circumstances as w^e have described ; but, when there is even a small 
degree of cover on the steam side, this plan of taking more than all 
the cover off the exhausting side ought never to be resorted to, aa 
it can serve no good purpose, and will materially increase an evil 
we have already explained, viz. the opening of the exhausting-port 
behind the piston before the stroke is nearly completed. The tables 
apply equally to the common short slide three-ported valves and to 
the long D valves. 

In fig. 1 is exhibited a common arrangement of the valves in \(h 




r2 



Fig. 1. 



Fig. 2. 



Fig. 3. 



234 THE PRACTICAL MODEL CALCULATOR. 

comotive engines, and in figs. 2 and 3 is shown an arrangement 
for working valves by a shifting cam, by which the amount of ex- 
pansion may be varied. This particular arrangement, however, is 
antiquated, and is now but little used. 

The extent to which expansion can be carried beneficially by 
means of lap upon the valve is about one-third of the stroke ; that 
is, the valve may be made with so much lap, that the steam will be 
cut off when one-third of the stroke has been performed, leaving 
the residue to be accomplished by the agency of the expanding 
steam ; but if more lap be put on than answers to this amount of 
expansion, a very distorted action of the valve will be produced, 
which will impair the efficiency of the engine. If a further amount 
of expansion than this is wanted, it may be accomplished by wire- 
drawing the steam, or by so contracting the steam passage, that 
the pressure within the cylinder must decline when the speed of 
the piston is accelerated, as it is about the middle of the stroke. 
Thus, for example, if the valve be so made as to shut off the steam 
by the time two-thirds of the stroke have been performed, and the 
steam be at the same time throttled in the steam pipe, the full 
pressure of the steam within the cylinder cannot be maintained ex- 
cept near the beginning of the stroke where the piston travels 
slowly ; for as the speed of the piston increases, the pressure neces- 
sarily subsides, until the piston approaches the other end of the 
cylinder, where the pressure would rise again but that the operation 
of the lap on the valve by this time has had the effect of closing 
the communication between the cylinder and steam pipe, so as to 
prevent more steam from entering. By throttling the steam, there- 
fore, in the manner here indicated, the amount of expansion due to 
the lap may be doubled, so that an engine with lap enough upon 
the valve to cut off the steam at two-thirds of the stroke, may, by 
the aid of wire-drawing, be virtually rendered capable of cutting 
off the steam at one-third of the stroke. The usual manner of cut- 
ting off the steam, however, is by means of a separate valve, termed 
an expansion valve ; but such a device appears to be hardly neces- 
sary in many engines. In the Cornish engines, where the steam 
is cut off in some cases at one-twelfth of the stroke, a separate valve 
for the admission of steam, other than that which permits its es- 
cape, is of course indispensable ; but in common rotative engines, 
which may realize expansive efficacy by throttling, a separate ex- 
pansive valve does not appear to be required. In all engines there 
is a point beyond which expansion cannot be carried with advantage, 
as the resistance to be surmounted by the engine will then become 
equal to the impelling power ; but in engines working with a high 
pressure of steam that point is not so speedily attained. 

In high pressure, as contrasted with condensing engines, there is 
always the loss of the vacuum, which will generally amount to 12 
or 13 lbs. on the square inch, and in high pressure engines there is 
a benefit arising from the use of a very high pressure over a pres- 
sure of a moderate account. In all high pressure engines, there is 



THE STEAM ENGINE. 



^ 



a diminution in tlie power caused by the counteracting pressure of 
the atmosphere on the educting side of the piston ; for the force 
of the piston in its descent would obviously be greater, if there was 
a vacuum beneath it ; and the counteracting pressure of the atmo- 
sphere is relatively less when the steam used is of a very high 
pressure. It is clear, that if we bring down the pressure of the 
steam in a high pressure engine to the pressure of the atmosphere, 
it will not exert any power at all, whatever quantity of steam may 
be expended, and if the pressure be brought nearly as low as that 
of the atmosphere, the engine will exert only a very small amount 
of power ; whereas, if a very high pressure be employed, the pres- 
sure of the atmosphere will become relatively as small in counter- 
acting the impelling pressure, as the attenuated vapour in the con- 
denser of a condensing engine is in resisting the lower pressure 
which is there employed. Setting aside loss from friction, and sup- 
posing the vacuum to be a perfect one, there would be no benefit 
arising from the use of steam of a high pressure in condensing en- 
gines, for the same weight of steam used without expansion, or 
with the same measure of expansion, would produce at every pres- 
sure the same amount of mechanical power. A piston with a 
square foot of area, and a stroke of three feet with a pressure of 
one atmosphere, would obviously lift the same weight through the 
same distance, as a cylinder with half a square foot of area, a stroke 
of three feet, and a pressure of two atmospheres. In the one case, 
we have three cubic feet of steam of the pressure of one atmosphere, 
and in the other case IJ cubic feet of the pressure of two atmo- 
spheres. But there is the same weight of steam, or the same quan- 
tity of heat and water in it, in both cases ; so that it appears a given 
weight of steam would, under such circumstances, produce a definite 
amount of power, without reference to the pressure. In the case 
of ordinary engines, however, these conditions do not exactly apply ; 
the vacuum is not a perfect one, and the pressure of the resisting 
vapour becomes relatively greater as the pressure of the steam is 
diminished ; the friction also becomes greater from the necessity 
of employing larger cylinders, so that even in the case of condensing 
engines, there is a benefit arising from the use of steam of a con- 
siderable pressure. Expansion cannot be carried beneficially to any 
great extent, unless the initial pressure be considerable ; for if steam 
of a low pressure were used, the ultimate tension would be reduced 
to a point so nearly approaching that of the vapour in the con- 
denser, that the difi'erence would not suffice to overcome the friction 
of the piston ; and a loss of power would be occasioned by carrying 
expansion to such an extent. In some of the Cornish engines, the 
steam is cut off at one-twelfth of the stroke ; but there would be a 
loss arising from carrying the expansion so far, instead of a gain, 
unless the pressure of the steam were considerable. It is clear, 
that in the case of engines which carry expansion very far, a very 
perfect vacuum in the condenser is more important than it is in 
other cases. Nothing can be easier than to compute the ultimate 



236 THE PRACTICAL MODEL CALCULATOR. 

pressure of expanded steam, so as to see at what point expansion 
ceases to be productive of benefit ; for as the pressure of expanded 
steam is inversely as the space occupied, the terminal pressure when 
the expansion is twelve times is just one-twelfth of what it was at 
first, and so on, in all other projections. The total pressure should 
be taken as the initial pressure — not the pressure on the safety 
valve, but that pressure plus the pressure of the atmosphere. 

In high pressure engines, working at from 70 to 90 lbs. on the 
square inch, as in the case of locomotives, the efficiency of a given 
quantity of water raised into steam may be considered to be about 
the same as in condensing engines. If the pressure of steam in a 
high pressure engine be 120 lbs., or 125 lbs. above the atmosphere, 
then the resistance occasioned by the atmosphere will cause a loss 
of |th of the power. If the pressure of the steam in a low pressure 
engine be 16 lbs. on the square inch, or 11 lbs. above the atmo- 
sphere, and the tension of the vapour in the condenser be equiva- 
lent to 4 inches of mercury, or 2 lbs. of pressure on the square 
inch, then the resistance occasioned by this rare vapour will also 
cause a loss of |th of the power. A high pressure engine, there- 
fore, with a pressure of 105 lbs. above the atmosphere, works with 
only the same loss from resistance to the piston, as a low pressure 
engine with a pressure of 1 lb. above the atmosphere, and with 
these proportions the power produced by a given weight of steam 
will be the same, whether the engine be high pressure or con- 
densing. 

SPHEROIDAL CONDITION OF WATER IN BOILERS. 

Some of the more prominent causes of boiler explosions have 
been already enumerated ; but explosions have in some cases been 
attributed to the spheroidal condition of the water in the boiler, 
consequent upon the flues becoming red-hot from a deficiency of 
water, the accumulation of scale, or otherwise. The attachment 
of scale, from its imperfect conducting power, will cause the iron 
to be unduly heated ; and if the scale be accidentally detached, a 
partial explosion may occur in consequence. It is found, that a 
sudden disengagement of steam does not immediately follow the 
contact of water with the hot metal, for water thrown upon red- 
hot iron is not immediately converted into steam, but assumes the 
spheroidal form and rolls about in globules over the surface. These 
globules, however high the temperature of the metal may be on 
which they are placed, never rise above the temperature of 205°, 
and give off but very little steam ; but if the temperature of the 
metal be lowered, the water ceases to retain the spheroidal form, 
and comes into intimate contact with the metal, whereby a rapid 
disengagement of steam takes place. If water be poured into a very 
hot copper flask, the flask may be corked up, as there will be scarce 
any steam produced so long as the high temperature is maintained; 
but so soon as the temperature is sufi"ered to fall below 350° or 
400°, the spheroidal condition being no longer maintainable, steam 
is generated with rapidity, and the cork will be projected from the 



THE STEAM ENGINE. 237 

mouth of the flask with great force. In a boiler, no doubt, where 
there is a considerable head of water, the repellant action of the 
spheroidal globules will be more effectually counteracted than in 
the small vessels employed in experimental researches. But it is 
doubtful whether in all boilers there may not be something of the 
spheroidal action perpetually in operation, and leading to effects at 
present mysterious or inexplicable. 

One of the most singular phenomena attending the spheroidal 
condition is, that the vapour arising from a spheroid is of a far 
higher temperature than the spheroid itself. Thus, if a thermometer 
be held in the atmosphere of vapour which surrounds a spheroid of 
water, the mercury, instead of standing at 205°, as would be the 
case if it had been immersed in the spheroid, will rise to a point 
determinable by the temperature of the vessel in which the spheroid 
exists. In the case of a spheroid, for example, existing within a 
crucible raised to a temperature of 400°, the thermometer, if held 
in the vapour, will rise to that point ; and if the crucible be made 
red-hot, the thermometer will be burst, from the boiling point of 
mercury having been exceeded. A part of this effect may, indeed, 
be traced to direct radiation, yet it appears indisputable, from the 
experiments which have been made, that the vapour of a liquid 
spheroid is much hotter than the spheroid itself. 

EXPANSION. 

At page 131 we have given a table of hyperbolic or Byrgean 
logarithms, for the purpose of facilitating computations upon this 
subject. 

Let the pressure of the steam in the boiler be expressed by unity, 
and let x represent the space through which the piston has moved 
whilst urged by the expanding steam. The density will then be 

, and, assuming that the densities and elasticities are pro- 

JL ~i~ X 

portionate, will be the differential of the efficiency, and the 

1 + X 

efficiency itself will be the integral of this, or, in other words, the 

hyperbolic logarithm of the denominator ; wherefore the efficiency 

of the whole stroke will be 1 + log. (1 + x). 

Supposing the pressure of the atmosphere to be 15 lbs., 15 -f 35 
= 50 lbs., and if the steam be cut off at |^th of the stroke, it will be 
expanded into four times its original volume ; so that at the ter- 
mination of the stroke, its pressure will be 50-r-4=12-2 lbs., or 2*8 
lbs. less than the atmospheric pressure. 

When the steam is cut off at one-fourth, it is evident that a? = 3. 
In such case the efficiency is 

1 + log. (1 -f 3), or 1 -f log. 4. 

The hyperbolic logarithm of 4 is 1*386294, so that the efficiency 
of the steam becomes 2-386294 ; that is, by cutting off the steam 
at ^, more than twice the effect is produced with the same consump- 
tion of fuel J in other words, one-half of the fuel is saved. 



238 THE PRACTICAL MODEL CALCULATOR. 

This result may thus be expressed in words : — Divide the length 
of the stroke through which the steam expands by the length of 
stroke performed with the full pressure, which last portion call 1 ; 
the hyperbolic logarithm of the quotient is the increase of efficiency 
due to expansion. We introduce on the following page more de- 
tailed tables, to facilitate the computation of the power of an en- 
gine working expansively, or rather to supersede the necessity of 
entering into a computation at all in each particular case. 

The first column in each of the following tables contains the 
initial pressure of the steam in pounds, and the remaining columns 
contain the mean pressure of steam throughout the stroke, with the 
different degrees of expansion indicated at the top of the columns, 
and which express the portion of the stroke during which the steam 
acts expansively. Thus, for example, if steam be admitted to the 
cylinder at a pressure of 3 pounds per square inch, and be cut off 
within Jth of the end of the stroke, the mean pressure during the 
whole stroke will be 2*96 pounds per square inch. In like manner, 
if steam at the pressure of 3 pounds per square inch were cut off 
after the piston had gone through |^th of the stroke, leaving the 
steam to expand through the remaining Jth, the mean pressure 
during the whole stroke would be 1-164 pounds per square inch. 

FRICTION. 

The friction of iron sliding upon brass, which has been oiled and 
then wiped dry, so that no film of oil is interposed, is about ^ of 
the pressure ; but in machines in actual operation, where there is a 
film of oil between the rubbing surfaces, the fraction is only about 
one-third of this amount, or ^d of the weight. The tractive re- 
sistance of locomotives at low speeds, which is entirely made up of 
friction, is in some cases g^th of the weight ; but on the average 
about g^th of the load, which nearly agrees with my former state- 
ment. If the total friction be g^th of the load, and the rolling 
friction be iwoth of the load, then the friction of attrition must be 
^th of the load ; and if the diameter of the wheels be 36 in., and the 
diameter of the axles be 3 in., which are common proportions, the 
friction of attrition must be increased in the proportion of 36 to 3, 
or 12 times, to represent the friction of the rubbing surface when 
moving with the velocity of the carriage, ^^ths are about J^th of 
the load, which does not differ much from the proportion of ^\d, as 
previously stated. While this, however, is the average result, the 
friction is a good deal less in some cases. Engineers, in some 
experiments upon the friction, found the friction to amount to 
less than ^th of the weight ; and in some experiments upon the 
friction of locomotive axles, it was found that by ample lubrication 
the friction might be made as little as ^th of the weight, and the 
traction, with the ordinary size of wheels, would in such a case be 
about 5^oth of the weight. The function of lubricating substances 
is to prevent the rubbing surfaces from coming into contact, where- 
by abrasion would be produced, and unguents are effectual in this 



THE STEAM ENGINE. 



23a 



EXPANDED STEAM. 



■MEAN PRESSURE AT DIFFERENT DENSITIES AND 
RATE OF EXPANSION. 



The column headed contains the initial p}'essu7'e in lbs., and the remaining columns 
cantain the mean pressure in lbs., with different grades of expansion. 



Expansion by Eighths. 





1 


2 


3 


4 


6 


6 


7 


8 


8 


8 


8 


8 


8 


8 


3 


2-96 


2-89 


2-75 


2-53 


2-22 


1-789 


1-154 


4 


3-95 


3-85 


3-67 


3-38 


2-96 


2-386 


1-539 


5 


4-948 


4-818 


4-593 


4-232 


3-708 


2-982 


1-924 


6 


5-937 


5-782 


5-512 


5-079 


4-450 


3-579 


2-309 


7 


6-927 


6-746 


6-431 


5-925 


5-241 


4-175 


2-694 


8 


7-917 


7-710 


7-350 


6-772 


5-934 


4-772 


3-079 


9 


8-906 


8-673 


8-268 


7-618 


6-675 


5-368 


3-463 


10 


9-896 


9-637 


9-187 


8-465 


7-417 


5-965 


3-848 


11 


10-885 


10-601 


10-106 


9-311 


8-159 


6-561 


4-233 


12 


11-875 


11-565 


10-925 


10-158 


8-901 


7-158 


4-618 


13 


12-865 


12-528 


11-943 


11-004 


9-642 


7-754 


5-003 


14 


13-854 


13-492 


12-862 


11-851 


10-384 


8-531 


5-388 


15 


14-844 


14-456 


13-781 


12-697 


11-126 


8-947 


5-773 


16 


15-834 


15-420 


14-700 


13-544 


11-868 


9-544 


6-158 


17 


16-823 


16-383 


15-618 


14-390 


12-609 


10-140 


6-542 


18 


17-813 


17-347 


16-537 


15-237 


13-351 


10-737 


6-927 


19 


18-702 


18-311 


17-448 


16-803 


14-093 


11-383 


7-312 


20 


19-792 


19-275 


18-375 


16-930 


14-835 


11-930 


7-697 


25 


24-740 


24-093 


22-968 


21-162 


18-543 


14-912 


9-621 


30 


29-688 


28-912 


27-562 


25-395 


22-252 


17-895 


11-546 


35 


34-636 


33-731 


33-156 


29-627 


25-961 


20-877 


13-470 


40 


39-585 


38-550 


36-750 


33-860 


29-670 


23-860 


15-395 


45 


44-533 


43-368 


41-343 


38-092 


33-378 


26-842 


17-319 


50 


49-481 


48-187 


45-937 


42-325 


37-067 


29-825 


19-243 



Expansion by Tenths. 





^, 


T% 


1^. 


^ 


^ 


^ 


^ 


^ 


1^ 


3 


2-980 


2-930 


2-830 


2-710 


2-639 


2-299 


1-981 


1-668 


0-990 


4 


3-974 


3-913 


3-780 


3-614 


3-386 


3-065 


2-642 


2-087 


1-320 


5 


4-968 


4-892 


4-725 


4-518 


4-232 


3-832 


3-303 


2-609 


1-651 


6 


5-961 


5-870 


5-670 


5-421 


5-079 


4-598 


3-963 


3-130 


1-981 


7 


6-955 


6-848 


6-615 


6-325 


5-925 


5-364 


4-624 


3-652 


2-311 


8 


7-948 


7-827 


7-560 


7-228 


6-772 


6-131 


5-284 


4-174 


2-641 


9 


8-942 


8-805 


8-505 


8-132 


7-618 


6-897 


6-945 


4-696 


2-971 


10 


9-936 


9-784 


9-450 


9-036 


8-465 


7-664 


6-606 


5-218 


3-302 


11 


10-929 


10-762 


10-395 


9-939 


9-311 


8-430 


7-266 


5-739 


3-632 


12 


11-923 


11-740 


11-340 


10-843 


10-158 


9-196 


7-927 


6-261 


3-962 


13 


12-856 


12-719 


12-285 


11-746 


10-994 


9-963 


8-587 


6-783 


4-292 


14 


13-910 


13-967 


13-230 


12-650 


11-851 


10-729 


9-248 


7-305 


4-622 


15 


14-904 


14-676 


14-175 


13-554 


12-697 


11-496 


9-909 


7-827 


4-953 


16 


15-897 


15-654 


15-120 


14-457 


13-544 


12-262 


10-569 


8-348 


5-283 


17 


16-891 


16-632 


16-065 


15-361 


14-051 


13-028 


11-230 


8-870 


5-613 


18 


17-884 


17-611 


17-010 


16-264 


15-237 


13-795 


11-890 


9-392 


5-944 


19 


18-878 


18-589 


17-955 


17-168 


16-083 


14-561 


12-551 


9-914 


6-273 


20 


19-872 


19-568 


18-900 


18-072 


16-930 


15-328 


13-212 


10-436 


6-600 


25 


24-840 


24-460 


23-625 


22-590 


21-162 


19-160 


16-515 


13-040 


8-255 


80 


29-808 


29-352 


28-350 


27-108 


25-395 


22-992 


19-818 


15-654 


9-906 


85 


34-776 


34-244 


33-075 


31-626 


29-627 


26-824 


23-121 


18-263 


11-557 


40 


39-744 


39-136 


37-800 


36-144 


33-860 


30-656 


26-224 


20-872 


13-208 


45 


44-912 


44-028 


42-525 


40-662 


38-092 


34-888 


29-727 


23-481 


14-859 


50 


49-680 


48-920 


47-250 


45-180 


42-325 


38-320 


33-030 


26-090 


16-510 



240 THE PRACTICAL MODEL CALCULATOR. 

respect in the proportion of their viscidity ; but if the viscidity of 
the unguent be greater than what suffices to keep the surfaces 
asunder, an additional resistance will be occasioned ; and the nature 
of the unguent selected should always have reference, therefore, to 
the size of the rubbing surfaces, or to the pressure per square inch 
upon them. With oil, the friction appears to be a minimum when 
the pressure on the surface of a bearing is about 90 lbs. per square 
inch : the friction from too small a surface increases twice as rapidly 
as the friction from too large a surface ; added to which, the bear- 
ing, when the surface is too small, wears rapidly away. For all 
sorts of machinery, the oil of Patrick Sarsfield Devlan, of Reading, 
Pa., is the best. 

HORSE POWER. 

A horse power is an amount of mechanical force capable of rais- 
ing 33,000 lbs. one foot high in a minute. The average force ex- 
erted by the strongest horses, amounting to 33,000 lbs., raised one 
foot high in the minute, was adopted, and has since been retained. 
The efficacy of engines of a given size, however, has been so much 
increased, that the dimensions answerable to a horse power then, 
will raise much more than 33,000 lbs. one foot high in the minute 
now ; so that an actual horse power, and a nominal horse power 
are no longer convertible terms. In some engines every nominal 
horse power will raise 52,000 lbs. one foot high in the minute, in 
others 60,000 lbs., and in others 66,000 lbs. ; so that an actual and 
nominal horse power are no longer comparable quantities, — the one 
being a unit of dimension, and the other a unit of force. The ac- 
tual horse power of an engine is ascertained by an instrument called 
an indicator ; but the nominal power is ascertained by a reference 
to the dimensions of the cylinder, and may be computed by the 
following rule : — Multiply the square of the diameter of the cylin- 
der in inches by the velocity of the piston in feet per minute, and 
divide the product by 6,000 ; the quotient is the number of nominal 
horses power. In using this rule, however, it is necessary to adopt 
the speed of piston which varies with the length of the stroke. The 
speed of piston with a two feet stroke is, according to this system, 
160 per minute ; with a 2 ft. 6 in. stroke, 170 ; 3 ft., 180 ; 3 ft., 6 
in., 189 ; 4 ft., 200 ; 5 ft., 215 ; 6 ft., 228 ; 7 ft., 245 ; 8 ft., 256 ft. 

By ascertaining the ratio in which the velocity of the piston 
increases with the length of the stroke, the element of velocity may 
be cast out altogether ; and this for most purposes is the most con- 
venient method of procedure. To ascertain the nominal power by 
this method, multiply the square of the diameter of the cylinder in 
inches by the cube root of the stroke in feet, and divide the pro- 
duct by 47 ; the quotient is the number of nominal horses power 
of the engine. This rule supposes a uniform effective pressure upon 
the piston of 7 lbs. per square inch ; the effective pressure upon 
the piston of 4 horse power engines of some of the best makers 
has been estimated at 6*8 lbs. per square inch, and the pressure 



THE STEAM ENGINE. 241 

increased slightly with the power, and became 6*94 lbs. per square 
inch in engines of 100 horse power; but it appears to be more con- 
venient to take a uniform pressure of 7 lbs. for all powers. Small 
engines, indeed, are somewhat less effective in proportion than large 
ones ; but the difference can be made up by slightly increasing the 
pressure in the boiler ; and small boilers will bear such an increase 
without inconvenience. 

Nominal power, it is clear, cannot be transformed into actual 
power, for the nominal horse power expresses the size of an engine, 
and the actual horse power the number of times 33,000 lbs. it will 
lift one foot high in a minute. To find the number of times 33,000 
lbs. or 528 cubic feet of water, an engine will raise one foot high 
in a minute, — or, in other words, the actual power, — we first find the 
pressure in the cylinder by means of the indicator, from which we 
deduct a pound and a half of pressure for friction, the loss of 
power in working the air pump, &c. ; multiply the area of the 
piston in square inches by this residual pressure, and by the motion 
of the piston, in feet per minute, and divide by 33,000; the 
quotient is the actual number of horse power. The same result is 
attained by squaring the diameter of the cylinder, multiplying by 
the pressure per square inch, as shown by the indicator, less a pound 
and a half, and by the motion of the piston in feet, and dividing by 
42,017. The quantity thus arrived at, will, in the case of nearly all 
modern engines, be very different from that obtained by multiplying 
the square of the diameter of the cylinder by the cube root of the 
stroke, and dividing by 47, which expresses the nominal power ; and 
the actual and nominal power must by no means be confounded, as 
they are totally different things. The duty of an engine is the 
work done in relation to the fuel consumed, and in ordinary mill or 
marine engines it can only be ascertained by the indicator, as the 
load upon such engines is variable, and cannot readily be deter- 
mined : but in the case of engines for pumping water, where the 
load is constant, the number of strokes performed by the engine 
represents the duty ; and a mechanism to register the number of 
strokes made by the engine in a given time, is a sufiicient test of 
the engine's performance. 

In high pressure engines the actual power is readily ascertained 
by the indicator, by the same process by which the actual power of 
low pressure engines is ascertained. The friction of a locomotive 
engine when unloaded, is found by experiment to be about 1 lb. per 
square inch on the surface of the pistons, and the additional friction 
caused by any additional resistance is estimated at about -14 of 
that resistance ; but it will be a sufficiently near approximation to 
the power consumed by friction in high pressure engines, if we 
make a deduction of a pound and a half from the pressure on that 
account, as in the case of low pressure engines. High pressure 
engines, it is true, have no air pump to work ; but the deduction of 
a pound and a half of pressure is relatively a much smaller one 
where the pressure is high than where it does not much exceed the 
V 16 



242 THE PRACTICAL MODEL CALCULATOR. 

pressure of the atmosphere. The rule, therefore, for the actual 
horse power of a high pressure engine will stand thus : — Square 
the diameter of the cylinder in inches, multiply by the pressure of 
the steam in the cylinder per square inch, less 1^ lbs., and by the 
speed of the piston in feet per minute, and divide by 42,017 ; the 
quotient is the actual horse power. The nominal horse power of a 
high pressure engine has never been defined ; but it should obvi- 
ously hold the same relation to the actual power as that which 
obtains in the case of condensing engines, so that an engine of a 
given nominal horse power may be capable of performing the same 
work, whether high pressure or condensing. This relation is main- 
tained in the following rule, which expresses the nominal horse 
power of high pressure engines : — Multiply the square of the diame- 
ter of the cylinder in inches by the pressure on the piston in pounds 
per square inch, and by the speed of the piston in feet per minute, 
and divide the product by 120,000 ; the quotient is the power of 
the engine in nominal horses power. If the pressure upon the 
piston be 80 lbs. per square inch, the operation may be abbreviated 
by multiplying the square of the diameter of the cylinder by the 
speed of the piston, and dividing by 1,500, which will give the 
same result. This rule for nominal horse power, however, is not 
representative of the dimensions of the cylinder ; but a rule for the 
nominal horse power of high pressure engines which shall discard 
altogether the element of velocity, is easily constructed ; and, as 
diflferent pressures are used in different engines, the pressure must 
become an element in the computation. The rule for the nominal 
power will therefore stand thus : — Multiply the square of the 
diameter of the cylinder in inches by the pressure on the piston in 
pounds per square inch, and the cube root of the stroke in feet, and 
divide the product by 940 ; the quotient is the power of the engine 
in nominal horse power, the engine working at the ordinary speed 
of 128 times the cube root of the stroke. 

A summary of the results arrived at by these rules is given in 
the following tables, which, for the convenience of reference, we 
introduce. 



PARALLEL MOTION. 

Rule I. — In such a combination of two levers as is represented in 
Figs. 1 and 2, page 245, to find the length of radius bar required 
for any given length of lever C (r, and proportion of parts of the link, 
Cr E and F E, so as to make the point E move in a perpendicular 
line. — Multiply the length of G C by the length of the segment G E, 
and divide the product by the length of the segment F E. The 
quotient is the length of the radius bar. 

Rule II. — {Fig. 2, page 245.) The length of the radius bar and 
of Q Gr being given, to find the length of the segment (FE) of the 
link next the radius bar. — Multiply the length of C G by the 



THE STEAM ENGINE. 



243 



Table of Nominal Eorse 


Power of Low Pressure Engines. 


III 


Length of Stroke in Feet. 


1 


1^ 


2 


^ 


3 


H 


4 


^ 


5 


5^ 


6 


7 


4 


•34 


•39 


•43 


•46 


•49 


•52 


•64 


•56 


•68 


•60 


•62 


•65 


5 


•53 


•61 


•67 


•72 


•76 


-81 


•84 


•88 


•91 


•94 


•96 


1-02 


6 


•76 


•87 


•96 


1-04 


1-10 


1-16 


1-22 


1-26 


1-31 


1-36 


1^39 


1-47 


7 


1-04 


1^19 


1^31 


1-41 


1-60 


1-58 


1-66 


1-72 


1-78 


1^84 


1-89 


1-99 


8 


1-36 


1-56 


1-72 


1-85 


1-96 


2-07 


2-16 


2-26 


2-33 


2-40 


2-47 


2-60 


9 


1-72 


1-97 


2-17 


2-34 


2-49 


2-62 


2-74 


2-84 


2-95 


3-04 


3-13 


3-30 


10 


213 


2-44 


2-68 


2-89 


3-07 


3-23 


3-38 


3-51 


3-64 


3-76 


3-87 


4-07 


11 


2-57 


2-95 


3-24 


3-49 


3-77 


3-91 


4-15 


4-25 


4-40 


4-54 


4-68 


4-92 


12 


3-06 


3-51 


3-86 


4-16 


4-42 


4-65 


4-86 


5-06 


5-24 


5-41 


6-57 


5-86 


13 


3-60 


4-12 


4-53 


4-88 


5^19 


5-46 


5-64 


5-94 


6-15 


6-35 


6-63 


6-88 


14 


4-17 


4-77 


5-25 


5-66 


6-01 


6-33 


6-62 


6-88 


7-13 


7-36 


7-68 


7-98 


15 


4-77 


5-48 


6-03 


6-50 


6-90 


7-27 


7-60 


7-90 


8-19 


8-45 


8-70 


9-16 


16 


5-45 


6-23 


6-86 


7-39 


7-86 


8-27 


8-65 


8-99 


9-31 


9-61 


9-90 


10-42 


17 


615 


7-04 


7-75 


8-35 


8-86 


9^34 


9-76 


10-15 


10-52 


10-86 


11-17 


11-76 


18 


6-89 


7-89 


8-68 


9-36 


9-94 


10-47 


10-94 


11-38 


11-79 


12-17 


12-53 


13-19 


19 


7-68 


8-79 


9-68 


10-42 


11-17 


11-66 


12-19 


12-68 


13-13 


13-56 


13-96 


14-69 


20 


8-51 


9-74 


10-72 


11-55 


12-27 


12-92 


13-51 


14-05 


14-55 


15-02 


15-46 


16-28 


22 


10-30 


11-79 


12-97 


13-98 


14-85 


15-63 


16-62 


17-30 


17-65 


18-18 


18-71 


19-70 


24 


12-26 


1403 


16-44 


16-63 


17-67 


18-61 


19-45 


20-23 


20^95 


31-63 


22-27 


23-44 


26 


14-39 


16-46 


18-12 


19-52 


20-75 


21-84 


22-56 


23-75 


24-6 


25-39 


26-14 


27-61 


28 


16-C8 


19-09 


21-02 


22-64 


24-06 


25-33 


26-48 


27-54 


28-62 


29-44 


30-31 


31-90 


30 


19-15 


21-92 


24-13 


25-99 


27-62 


29-07 


30-40 


31-61 


32-74 


33-80 


34-80 


36-63 


32 


21-79 


24-96 


27-51 


29-67 


31-42 


33-08 


34-69 


35-97 


37-26 


38-46 


39-59 


41-68 


34 


24-60 


28-16 


30-99 


33-39 


35-44 


37-34 


39-04 


40-60 


4206 


43-41 


44-69 


47 06 


36 


27-57 


31-56 


34-74 


37-42 


39-77 


41-87 


43-77 


45-52 


47-15 


48-67 


50-11 


62-75 


38 


30-72 


35-17 


38-71 


41-69 


44-66 


46-64 


48-77 


50-72 


52-54 


54-23 


55-83 


68-78 


40 


34-04 


38-97 


42-89 


46-20 


49-10 


51-69 


54-04 


56-20 


58-21 


60-09 


61-86 


66-12 


42 


37-53 


42-96 


47-29 


50-94 


54-13 


56-98 


59-58 


61-96 


64-18 


66-25 


68-21 


71-78 


44 


41-19 


47-15 


51-90 


56-91 


59-38 


62-54 


66-46 


68-00 


70^44 


72-71 


74-85 


78-79 


46 


45-02 


51-54 


56-72 


61-10 


64-88 


68-19 


71-43 


74-33 


76-69 


79-47 


81-81 


86-12 


48 


49-02 


56-11 


61-76 


66-58 


70-70 


74-42 


77-82 


80-94 


83-83 


86-53 


8908 


93-78 


50 


5319 


60-89 


67-02 


72-19 


76-71 


80-76 


84-44 


87-82 


90-96 


93-89 


96-66 


101-7 


62 


57-55 


65-86 


72-48 


78-08 


83-00 


87-35 


90-25 


94-98 


98-40 


101-55 


104-5 


110-0 


54 


62-04 


71-02 


78-17 


84-20 


89-48 


94-20 


98-49 


102-4 


106-1 


109-6 


112-7 


118-7 


56 


66-72 


76-38 


84-07 


90-55 


96-23 


101-30 


105-9 


110-1 


1141 


117-8 


121-2 


127-6 


58 


71-58 


81-93 


90-18 


97-14 


103-2 


108-6 


113-6 


118-2 


122-4 


126-3 


129-2 


136^7 


60 


76-60 


87-68 


96-50 


103-9 


110-4 


116-3 


121-6 


126-4 


131-0 


135-2 


139-2 


146-5 


62 


81-79 


93-62 


103-04 


111-0 


117-96 


124-18 


129-81 


13503 


139-86 


144-37 


148-6 


156-7 


64 


87-15 


99-84 


110-0 


118-3 


125-7 


132-3 


138-3 


143-9 


149-0 


153-82 


158-4 


166-7 


66 


92-68 


106-1 


116-8 


125-8 


133-6 


140-7 


147-3 


1530 


158-5 


163-6 


168-4 


177-3 


68 


98-40 


112-6 


123-9 


133-6 


141-8 


149-4 


156-2 


162-4 


168-2 


173-6 


178-8 


188-2 


70 


104-26 


119-3 


131-3 


141-6 


150-4 


158-3 


165-5 


172-1 


178-2 


1840 


189-4 


199-4 


72 


110-30 


126-2 


139-0 


149-7 


159-1 


167-4 


175-1 


182-1 


188-6 


194-7 


200-4 


211-0 


74 


116-5 


133-4 


146-8 


158-1 


167-9 


176-7 


185-4 


192-4 


199-2 


205-7 


211-6 


223-4 


76 


12-2-9 


140-7 


154-8 


166-8 


178-6 


186-6 


195-0 


202-9 


210-1 


216-9 


223-3 


235-1 


78 


129-4 


148-2 


163-1 


175-6 


186-7 


196-5 


205-4 


2121 


221-4 


228-6 


235-2 


247-6 


80 


136-2 


155-8 


171-6 


184-8 


196-4 


206-7 


2161 


224-8 


232-8 


240-4 


247-4 


260-5 


82 


143-0 


163-8 


180-2 


194-2 


206-2 


217-3 


226-9 


237-8 


244-6 


252-5 


2600 


273^8 


84 


1501 


171-8 


189-1 


203-8 


216-5 


227-9 


238-3 


247-8 


256-7 


265-0 


272-8 


287^1 


86 


157-4 


180-1 


198-2 


213-6 


227-0 


237-8 


247-4 


258-2 


269-1 


277-8 


286-0 


301-0 


88 


164-8 


188-6 


207-6 


223-6 


237-5 


250-2 


261-6 


272-0 


281-7 


290-8 


299-4 


316-2 


90 


172-3 


197-3 


•217-1 


233-9 


248-6 


261-7 


273-6 


284-5 


291-7 


304-2 


313-2 


329-7 



length of the link G F, and divide the product by the sum of the 
lengths of the radius bar and of C G. The quotient is the length 
required. 

Rule III. — (Figs. 3 and 4, pages 246 and 247.) To find the length 
of the radius bar (Fff), the length of Q Gr being given. — Square the 
length of C G, and divide it by the length of D G. The quotient 
is the length required. 

Rule IV. — {Figs. 3 and 4, pages 246 and 247.) To find the length 
of the radius bar, the horizontal distance of its centre (H) from the 
main centre being given. — To this given horizontal distance, add half 
the versed sine (D N) of the arc described by the end of beaifi (D). 
Square this sum. Take the same sum, and add to it the length of 



244 



THE PRACTICAL MODEL CALCULATOR. 



Table of Nominal Horse Power of High Pressure Engines. 



•S5 

ii 

5^" 


Length of Stroke in Feet. . 


1 


n 


2 


^ 


3 


3^ 


4 


^ 


5 


H 


6 


7 


2 


•25 


•29 


•32 


•35 


•37 


•38 


•40 


•42 


•44 


•45 


•46 


•49 


2i 


•39 


•45 


•50 


•54 


•57 


•60 


•63 


•06 


•68 


•70 


•72 


•76 


8 


•57 


•65 


•72 


•78 


•83 


•87 


•91 


•95 


•98 


1-01 


1-04 


1-10 


3i 


•78 


•89 


•98 


1-06 


1-13 


1-19 


1-24 


1-29 


1-34 


1-38 


1-42 


1-49 


4 


1-02 


1-17 


1-29 


1-38 


1-47 


1-56 


1-62 


1-68 


1-74 


1-80 


1-86 


1-95 


4i 


1-29 


1-48 


1-63 


1-75 


1-86 


1-96 


2-05 


2-13 


2-21 


2-28 


2-35 


2-47 


5 


1-59 


1-83 


2-01 


2-16 


2-28 


2-43 


2-52 


2-64 


2-73 


2-82 


2-88 


3-06 


5i 


1-93 


2-21 


2-43 


2-62 


2-78 


2-93 


3-12 


3-18 


3-50 


3-42 


3-51 


3-69 


6 


2-28 


2-61 


2-88 


3-12 


3-30 


3-48 


3-66 


3-78 


3-93 


4-05 


4-17 


4-41 


6i 


2-69 


3-09 


3-39 


3-66 


3-90 


4-08 


4-23 


4-44 


4-62 


4-77 


4-89 


6-16 


7 


312 


3-57 


3-93 


4-23 


4-50 


4-74 


4-95 


5-16 


5-34 


5-52 


6-67 


5-97 


7i 


3-CO 


4-11 


4-53 


4-86 


5-19 


5-46 


5-70 


6-94 


6-15 


6-33 


6-51 


6-87 


8 


4-08 


468 


5-16 


5-55 


5-88 


6-21 


6-48 


6-75 


6-99 


7-20 


7-41 


7-80 


8i 


4-62 


5-28 


6-82 


6-27 


6-63 


6-99 


7-32 


7-62 


7-89 


8-13 


8-37 


8-82 


9 


5-16 


5-91 


6-51 


7-02 


7-47 


7-86 


8-22 


8-52 


8-85 


9-12 


9-39 


9-90 


9i 


5-76 


6-60 


7-26 


7-80 


8-37 


8-76 


9-15 


9-51 


9-84 


10-17 


10-47 


10-01 


10 


6-39 


7-32 


8-04 


8-67 


9-21 


9-69 


10-14 


10-53 


10-92 


11-28 


11-61 


12-21 


lOi 


7-05 


8-04 


8-88 


9-54 


10-14 


10-68 


11-16 


11-61 


12-03 


12-42 


12-78 


13-47 


11 


7-71 


8-85 


9-72 


10-47 


11-31 


11-73 


12-45 


1275 


13-20 


13-62 


14-04 


14-76 


Hi 


8-43 


9-66 


10-62 


11-46 


12-15 


12-78 


13-80 


13-92 


14-61 


14-91 


15-33 


16-14 


12 


9-18 


10-53 


11-58 


12-41 


13-26 


13-95 


14-58 


15-18 


15-72 


16-23 


16-71 


17-58 


12i 


9-96 


11-40 


12-57 


13-53 


14-37 


15-15 


15-84 


16-47 


17-04 


17-58 


18-12 


19-08 


13 


10-80 


12-36 


13-59 


14-64 


15-57 


16-38 


16-92 


17-82 


18-45 


19-05 


19-69 


21-64 


13i 


11-64 


13-32 


14-64 


15-78 


16-77 


17-67 


38-48 


19-20 


19-89 


20-52 


21-16 


22-26 


14 


12-51 


14-31 


15-75 


16-98 


18-03 


18-99 


19-86 


20-64 


21-39 


22-08 


22-74 


23-94 


Mi 


13-41 


15-36 


16-92 


18-21 


19-35 


20-37 


21-30 


22-14 


22-95 


23-70 


24-39 


25-62 


15 


14-31 


16-44 


18-09 


19-50 


20-70 


21-81 


22-80 


23-70 


24-57 


25-35 


26-10 


27-48 


16 


16-35 


18-69 


20-58 


22-17 


23-58 


24-81 


25-95 


26-97 


27-93 


28-83 


29-70 


31-26 


17 


18-45 


21-12 


23-25 


25-05 


26-58 


2S-02 


29-28 


30-45 


31-56 


32-55 


33-57 


35-28 


18 


20-67 


23-67 


26-04 


28-08 


29-82 


31-41 


32-82 


34-14 


35-37 


36-51 


37-59 


39-57 


19 


23-04 


26-37 


29-04 


31-26 


33-51 


34-98 


36-57 


38-04 


39-39 


40-68 


41-88 


44-07 


20 


25-53 


29-22 


3-216 


34-65 


36-81 


38-76 


40-53 


42-16 


43-65 


45-06 


46-38 


48-84 


22 


30-90 


35-37 


38-91 


41-94 


44-55 


46-89 


49-86 


51-90 


52-96 


54-54 


56-13 


59-10 


24 


36-78 


42-09 


46-32 


49-89 


63-01 


55-83 


58-35 


60-69 


62-85 


64-89 


66-81 


70-32 


26 


43-17 


49-38 


54-36 


58-56 


62-25 


65-52 


67-68 


71-25 


73-80 


76-17 


78-42 


82-53 


28 


50-04 


57-27 


63-06 


67-92 


72-18 


75-99 


79-44 


82-62 


85-56 


88-32 


90-93 


95-70 


30 


57-45 


65-76 


72-39 


77-97 


82-86 


87-21 


91-20 


94-83 


98-22 


101-40 


104-4 


109-9 


32 


65-37 


74-88 


82-53 


88-71 


94-26 


99-24 


103-7 


107-9 


111-8 


115-4 


118-7 


125-0 


34 


73-80 


84-48 


92-9 


100-22 


106-3 


112-0 


117-1 


121-8 


126-2 


130-2 


134-0 


1411 


36 


82-71 


94-68 


104-2 


112-2 


119-3 


125-6 


131-3 


136-5 


141-4 


146-0 


160-3 


158-2 


38 


92-16 


105-5 


116-1 


125-0 


1340 


136-9 


146-3 


152-1 


157-6 


162-7 


167-6 


176-3 


40 


102-1 


116-9 


129-6 


128-6 


147-3 


155-1 


162-1 


168-6 


174-6 


180-2 


185-6 


195-3 


42 


112-6 


128-9 


141-8 


152-8 


162-4 


170-9 


178-7 


185-9 


192-5 


198-7 


204-6 


216-3 


44 


123-5 


141-4 


155-7 


167-7 


178-1 


187-6 


199-4 


204-0 


211-3 


218-1 


224-5 


236-3 


46 


135-0 


154-6 


170-1 


183-3 


194-6 


204-6 


214-3 


223-0 


230-0 


238-4 


246-4 


258-3 


48 


147-0 


168-3 


185-3 


199-6 


212-1 


223-2 


233-4 


242-8 


251-5 


259-6 


267-2 


281-3 


50 


159-6 


182-6 


201-0 


216-5 


230-1 


242-3 


253-3 


263-4 


272-9 


281-6 


289-9 


305-1 


52 


172-6 


197-6 


217-4 


234-2 


249-0 


262-0 


270-7 


284-9 


295-2 


304-6 


313-5 


330-0 


54 


186-1 


213-0 


234-5 


252-6 


268-4 


282-6 


295-4 


307-2 


318-3 


328-5 


338-1 


356-1 


56 


200-1 


2-29-1 


252-2 


271-6 


288-7 


303-9 


317-7 


330.3 


342-3 


353-4 


363-6 


382-8 


58 


214-7 


245-8 


270-5 


291-4 


309-6 


325-8 


340-8 


354-6 


367-2 


378-9 


389-7 


410-1 


60 


229-8 


268-0 


289-5 


311-7 


331-2 


348-9 


364-8 


379-2 


393-0 


405-6 


417-6 


439-5 



the beam (C D). Divide the square previously found bj this last 
sum, and the quotient is the length sought. 

Rule V. — [Figs. 5 and 6, pages 247, 248.) — To find the length 
of the radius har, C G and P Q being given. — Square C G, and 
multiply the square by the length of the side rod (P D) : call this 
product A. Multiply Q D by the length of the side lever (C D). 
From this product subtract the product of D P into C Gr, and divide 
A by the remainder. The quotient is the length required. 

Rule Nl.—[Figs. 5 and 6, pages 247, 248.) To find the length of 
the radius bar ; P Q, a7id the horizontal distance of the centre Hof 
the radius bar from the main centre being given. — To the given hori- 
zontal distance add half the versed sine (D N) of the arc described 



THE STEAM ENGINE. 



245 



h C^~^ 




Fig. 2. 




by tlie extremity (D) of the side lever. Square this sum and mul- 
tiply the square by the length of the side rod (P D). Call this pro- 
duct A. Take the same horizontal distance as before added to the 
same half versed sine (D N), and multiply the sum by the length of 
the side rod (P D) : to the product add the product of the length of 



v2 



246 



THE PRACTICAL MODEL CALCULATOR. 



Fig. 3. 




the side lever C D into the length of Q D, and divide A by the 
sum. The quotient will be the length required. 

When the centre H of the radius has its position determined, 
rules 4 and 6 will always give the length of the radius bar F H. 
To get the length of C G-, it will only be necessary to draw through 
the point F a line parallel to the side rod D P, and the point where 
that line cuts D C will be the position of the pin G. 

In using these formulas and rules, the dimensions must all be 
taken in the same measure ; that is, either all in feet, or all in 
inches ; and when great accuracy is required, the corrections given 
in Table (A) must be added to or subtracted from the calculated 
length of the radius bar, according as it is less or greater than the 
length of C Gr, the part of the beam that works it. 

1, Rule 4. — Let the horizontal distance (M C) of the centre (H) 



THE STEAM ENGINE. 
Fig. 4. 



247 



-^ 



# 



y 






x' E' 



Fig. 6. 



X 



-i.,...>; 




of the radius bar from the main centre be equal to 51 inches ; the 
half versed sine D N = 3 inches, and D C = 126 inches ; then by 
the rule we will have 

(51 + 3)2 (54)2 2916 _^ . , 

51 + 3 4- 126 ~ 180 ~ 180 - ^^ ^ ^^^^^^^ 
which is the required length of the radius bar (F H). 



248 



THE PRACTICAL MODEL CALCULATOR. 

Fig. 6. 



ly'X 




2. Rule 5. — The following dimensions are those of the Red Rover 
steamer: CG = 32 D P = 94 QD = 74 CD = 65 P Q = 20. 

By the rule we have, A = (32)2 x 94 = 96256 and 

96256 96256 

74 X 65 - 94 X 32 ~" 1802 ~ ^^'^' 
which is the required length of the radius bar. 

3. Rule 6. — Take the same data as in the last example, only- 
supposing that C G is not given, and that the centre H is fixed at 
a horizontal distance from the main centre, equal to 88*5 inches. 
Then the half versed sine of the arc D' D D" will be about 2 
inches, and we will have by the rule 

A = (83-5 + 2)2 X 94 = 705963-5 and 

A _ 705963-5 _ . 

85-5 X 94 + 65 X 74 ~ 1284-7 ~ ^^'^ ^^^^^^' 
the required length of the radius bar in this case. 

Table (A). 



F H 

This colnniTi gives ^^-^ when 

p p 

CG is the greater, and -— 

when F H is the greater. 


Correction to be added to or 
subtracted from the calcu- 
lated length of the radius 
bar, in decimal parts of its 
calculated length. 


1-0 
•9 
•8 
•7 
•6 
•5 
•4 



•0034 
•0075 
•0163 
•0270 
•0452 
•0817 



THE STEAM ENGINE. 249 

P P 

In both of the last two examples tt^ = '6 nearly. The correc- 
tion found by Table (A), therefore, would be 54 X -027 = 1*458 
inches, which must be subtracted from the lengths already found 
for the radius bar, because it is longer than C Gr. The corrected 
lengths will therefore be 

In example 2 F H = 51-94 inches. 

In example 3 F H = 53-34 inches. 

Rule. — To find the depth of the main beam at the centre. — Divide 
the length in inches from the centre of motion to the point where 
the piston rod is attached, by the diameter of the cylinder in 
inches ; multiply the quotient by the maximum pressure in pounds 
per square inch of the steam in the boiler ; divide the product by 
202 for cast iron, and 236 for malleable iron : in either case, the 
cube root of the quotient multiplied by the diameter of the cylinder 
in inches gives the depth in inches of the beam at the centre of 
motion. To find the breadth at the centre. — Divide the depth in 
inches by 16 ; the quotient is the breadth in inches. 

An engine beam is three times the diameter of the cylinder, from 
the centre to the point where the piston rod acts on it ; the force 
of the steam in the boiler when about to force open the safety 
valve is 10 lbs. per square inch. Required the depth and breadth 
when the beam is of cast iron. 

In this case w = 3, and P = 10, and therefore 

I 202 / 

The breadth = ^ D = -03 D. 
16 

It will be observed that our rule gives the least value to the 
depth. In actual practice, however, it is necessary to make allow- 
ance for accidents, or for faultiness in the materials. This may be 
done by making the depth greater than that determined by the 
rule ; or, perhaps more properly, by taking the pressure of the steam 
much greater than it can ever possibly be. As for the dimensions 
of the other parts of the beam, it is obvious that they ought to 
diminish towards the extremities ; for the power of a beam to resist 
a cross strain varies inversely as its length. The dimensions may 
be determined from the formula/ 5 cZ^ = 6 W ?. 

To apply the formula to cranks, we may assume the depth at the 
shaft to be equal to n times the diameter of the shaft ; hence, if 
w X D be the diameter of the shaft, the depth of the crank will 
be 71 X m X D. Substituting this in the formula fbd^ = 6 W /, 
and it becomes fb X n^ X m^ X D^ = 6 W I. Now, as before, 
W = -7854 X P X D^, so that the formula becomes fxbxn^x 
rr? — 4-7124 X P X ?. The value of n is arbitrary. In practice 
it may be made equal to IJ or 1-5. Taking this value, then, for 



250 THE PRACTICAL MODEL CALCULATOR. 

cast iron, the formula becomes 15300 x 5 X | X m^ = 4-7124 X 
P X Z, or 7305 m^h — V l\ but if L denote the length of the crank 
in feet, the formula becomes 609 nf 5 = PL, and .*. 6 = P X 
L -T- 609 m^. This formula may be put into the form of a rule, 
thus : — 

Rule. — To find the hreadtli at the shaft when the depth is equal 
to 1^ times the diameter of the shaft. — Divide the square of the 
diameter of the shaft in inches by the square of the diameter of 
the cylinder ; multiply the quotient by 609, and reserve the pro- 
duct for a divisor ; multiply the greatest elastic force of the steam 
in lbs. per square inch by the length of the crank in feet, and 
divide the product by the reserved divisor: the quotient is the 
breadth of the crank at the shaft. 

A crank shaft is \ the diameter of the cylinder ; the greatest 
possible force of the steam in the boiler is 20 lbs. per square inch ; 
and the length of the shaft is 3 feet. Required the breadth of the 
crank at the shaft when its depth is equal to 1 J times the diameter 
of the shaft. 

609 

In this case m = J, so that the reserved divisor — y^ = 38 : 

again, elastic force of steam in lbs. per square inch = 20 lbs. ; 

3 X 20 
hence width of crank = — qo — = 1*6 inches nearly. 

Rule. — To find the diameter of a revolving shaft. — Form a 
reserved divisor thus : multiply the number of revolutions which 
the shaft makes for each double stroke of the piston by the number 
1222 for cast iron, and the number 1376 for malleable iron. Then 
divide the radius of the crank, or the radius of the wheel, by the 
diameter of the cylinder ; multiply the quotient by the greatest 
pressure of the steam in the boiler expressed in lbs. per square 
inch ; divide the product by the reserved divisor ; extract the cube 
root of the quotient, and multiply the result by the diameter of 
the cylinder in inches. The product is the diameter of the shaft 
in inches. 

STRENGTH OF RODS WHEN THE STRAIN IS WHOLLY TENSILE ; SUCH AS 
THE PISTON ROD OF SINGLE ACTING ENGINES, PUMP RODS, ETC. 

Rule. — To find the diameter of a rod exposed to a tensile force 
only. — Multiply the diameter of the piston in inches by the square 
root of the greatest elastic force of the steam in the boiler esti- 
mated in lbs. per square inch ; the product, divided by 95, is the 
diameter of the rod in inches. 

Required the diameter of the transverse section of a piston rod 
in a single acting engine, when the diameter of the cylinder is 50 
inches, and the greatest possible force of the steam in the boiler is 
16 lbs. per square inch. Here, according to the formula, 
50 __ 200 
95 ^ ^5" ~ ^^ inches. 



THE STEAM ENGINE. 251 

Rule. — To find the strength of rods alternately extended and 
compressed, such as the piston rods of double acting engines. — Mul- 
tiply the diameter of the piston in inches by the square root of the 
maximum pressure of the steam in lbs. per square inch ; divide the 
product by 

47 for cast iron, 

50 for malleable iron. 

This rule applies to the piston rods of double acting engines, 
parallel motion rods, air-pump and force-pump rods, and the like. 
The rule may also be applied to determine the strength of connect- 
ing rods, by taking, instead of P, a number P', such that P' X sine 
of the greatest angle which the connecting rod makes with the 
direction = P. 

Supposing the greatest force of the steam in the boiler to be 16 
lbs. per square inch, and the diameter of the cylinder 50 inches ; 
required the diameter of the piston rod, supposing the engine to be 
double acting. In this case 

for cast iron c^ = -_ ^Z P = — _ — = 5 inches nearly ; 

47 47 ^ 

for malleable iron d = — \/ P = 4 inches. 

The pressure, however, is always taken in practice at more than 16 
lbs. If the pressure be taken at 25 lbs., the diameter of a malle- 
able iron piston rod will be 5 inches, which is the usual proportion. 
Piston rods are never made of cast iron, but air-pump rods are 
sometimes made of brass, and the connecting rods of land engines 
are cast iron in most cases. 

FORMULAS FOR THE STRENGTH OF VARIOUS PARTS OF MARINE ENGINES. 

The following general rules give the dimensions proper for the 
parts of marine engines, and we shall recapitulate, with all possible 
brevity, the data upon which the denominations rest. 

Let pressure of the steam in boiler = p lbs. per square inch, 
Diameter of cylinder = D inches. 
Length of stroke = 2 II inches. 

The vacuum below the piston is never complete, so that there 
always remains a vapour of steam possessing a certain elasticity. 
We may suppose this vapour to be able to balance the weight of the 
piston. Hence the entire pressure on the square inch of piston in 
lbs. = p + pressure of atmosphere = 15 + p. We shall substi- 
tute P for 15 + p. Hence 

Entire pressure on piston in lbs. = '7854 X (15 + p) x D^ 

= -7854 X P X D^. 

The dimensions of the paddle-shaft journal may be found from 
the following formulas, which are calculated so that the strain in 
ordinary working = | elastic force. 

Diameter of paddle-shaft journal = -08264 {R X P X D^js" 
Length of ditto = 1^ x diameter. 



252 THE PRACTICAL MODEL CALCULATOR. 

The dimensions of the several parts of the crank may be found 
from the following formulas, which are calculated so that the strain 
in ordinary working = one-half the elastic force ; and when one 
paddle is suddenly brought up, the strain at shaft end of crank = f 
elastic force, the strain at pin end of crank = elastic force. 

Exterior diameter of large eye = diameter of paddle-shaft + 
D [P X 1-561 X E^ -f -00494 x D^ x P^]* 



75-59 x^/il 

Length of ditto = diameter of paddle shaft. 

Exterior diameter of small eye = diameter of crank pin + 

•02521 X x/P X D. 

Length of ditto = -0375 X v7P X D. 
Thickness of web at paddle centre = 

D2 X P X ^/ {1-561 X R^ -j- -00494 x D^ x P} ' * 
9000 
Breadth of ditto = 2 X thickness. 
Thickness of web at pin centre — -022 X v^"P X D. 
Breadth of ditto = | X thickness. 

As these formulas are rather complicated, we may show what 
they become when ^ = 10 or P = 25. 

Exterior diameter of large eye = diameter of paddle shaft -f 



D5| 



Ds/ (1-561 X R^ -f -1235 x 

15-12 X ^/R 

Length of ditto = diameter of paddle shaft. 

Exterior diameter of small eye = equal diameter of crank pin -f 
•126 X D. 

Length of ditto = -1875 X B. 

Thickness of web at pin centre = -11 X D. 

Breadth of ditto = | X thickness of web. 

The dimensions of the crank pin journal may be found from the 
following formulas, which are calculated so that strain when bear- 
ing at outer end = elastic force, and in ordinary working strain = 
one-third of elastic force. 

Diameter of crank-pin journal = -02836 X ^/ P x D. 

Length of ditto = I x diameter. 

The dimensions of the several parts of the cross head may be found 
from the following formulas, in which we have assumed, for the 
purpose of calculation, the length = 1-4 X D. The formulas 

have been calculated so as to give the strain of web = x 

^ 2-225 

elastic force ; strain of journal in ordinary working = X elastic 

force, and when bearing at outer end = x elastic force. 

1-165 



THE STEAM ENGINE. 253 

Exterior diameter of eye = diameter of hole + -02827 X P^ x D. 

Depth of ditto = -0979 x P* x D^_ 
Diameter of journal = -01716 X v/ P X D. 
Length of ditto = | diameter of journal. 

Thickness of web at middle = -0245 X P* X D. 

Breadth of ditto = -09178 x P*" X D. 

Thickness of web at journal = -0122 X P^ x D. 

Breadth of ditto = -0203 x P^ x D. 

The dimensions of the several parts of the piston rod may be 
found from the following formulas, which are calculated so that the 
strain of piston rod = 4- elastic force. 

Diameter of the piston rod = -— -* 

50 

Length of part in piston = -04 X D X P. 

Major diameter of part in crosshead = -019 X -v/P X D. 

Minor diameter of ditto = -018 X >/P x D. 

Major diameter of part in piston = -028 X -v/P X D. 

Minor diameter of ditto = -023 X v/P x D. 

Depth of gibs and cutter through crosshead = -0358 X P^ X D. 

Thickness of ditto = -007 X P^" x D. _ 

Depth of cutter through piston = -017 X v/P X D. 

Thickness of ditto = -007 x P* x D. 

The dimensions of the several parts of the connecting rod may 
be found from the following formulas, which are calculated so that 
the strain of the connecting rod and the strain of the strap are both 
equal to one- sixth of the elastic force. j^ 

Diameter of connecting rod at ends = -019 X P^ x D. 

Diameter of ditto at middle = {1 -f -0035 X length in inches} 
X -019 X v/P X D. 

Major diameter of part in crosstail = -0196 X P^ x D. 

Minor ditto = -018 x P* x D. 

Breadth of butt = -0313 x P^ x D. 

Thickness of ditto = -025 x P* x D. 

Mean thickness of strap at cutter = -00854 x v/P x D. 

Ditto above cutter = -00634 x v/P x D. 

Distance of cutter from end of strap = -0097 X v/P X D. 

Breadth of gibs and cutter through crosstail = -0358 X P^ x D. 

1 
Breadth of gibs and cutter through butt = -022 X P^ x D. 

Thickness of ditto = -00564 x P^ x D. 
W 



254 THE PRACTICAL MODEL CALCULATOR. 

The dimensions of the several parts of the side rods may be 
found from the following formulas, which are calculated so as to 
make the strain of the side rod = one-sixth of elastic force, and 
the strains of strap and cutter = one-fifth of elastic force. 

Diameter of cylinder side rods at ends = -0129 x P^ X D. 
Diameter of ditto at middle = (1 -f '0035 X length in inches). 

X -0129 X pi X D. 

Breadth of butt = -0154 X P^ x D. 

Thickness of ditto = -0122 X P* X D. 

Diameter of journal at top end of side rod = '01716 X P^ X D. 
Length of journal at top end = | diameter. 

Diameter of journal at bottom end = '014 X P^ X D. 

Length of ditto = -0152 x P* X D. 

Mean thickness of strap at cutter = -00643 X P^ x D. 

Ditto below cutter = -0047 X P^ x D. 

Breadth of gibs and cutter = -016 X P^ x D. 

Thickness of ditto = -0033 x P^ x D. 

The dimensions of the main centre journal may be found from 
the following formulas, which are calculated so as to make the 
strain in ordinary working = one half elastic force. 

Diameter of main centre journal = -0367 X P^ X D. 
Length of ditto = | X diameter. 

The dimensions of the several parts of the air-pump may be 
found from the corresponding formulas given above, by taking for 
D another number d the diameter of air-pump. 

DIMENSIONS OF THE SEVERAL PARTS OF FURNACES AND BOILERS. 

Perhaps in none of the parts of a steam engine does the practice 
of engineers vary more than in those connected with furnaces and 
boilers. There are, no doubt, certain proportions for these, as well 
as for the others, which produce the maximum amount of useful 
effect for particular given purposes ; but the determination of these 
proportions, from theoretical considerations, has hitherto been at- 
tended with insuperable difficulties, arising principally from our im- 
perfect knowledge of the laws of combustion of fuel, and of the laws 
according to which caloric is imparted to the water in the boiler. 
In giving, therefore, the following proportions for the different 
parts, we desire to have it understood that we do not affirm them 
to be the best, absolutely considered ; we give them only as the 
average practice of the best modern constructors. In most of the 
cases we have given the average value per nominal horse power. 
It is well known that the term horse power is a conventional unit 
for measuring the size of steam engines, just as a foot or a mile is 



THE STEAM ENGINE. 255 

a unit for the measurement of extension. There is this difference, 
however, in the two cases, that whereas the length of a foot is 
fixed definitively, and is known to every one, the dimensions proper 
to an engine horse power differ in the practice of every different 
maker : and the same kind of confusion is thereby introduced into 
engineering as if one person were to make his foot-rule eleven 
inches long, and another thirteen inches. It signifies very little 
what a horse power is defined to be ; but when once defined, the 
measurement should be kept inviolable. The question now arises, 
what standard ought to be the accepted one. For our present pur- 
pose, it is necessary to connect by a formula the three quantities, 
nominal horses power, length of stroke, and diameter of cylinder. 
With this intention, 

Let S = length of stroke in feet, 

d = diameter of cylinder in inches ; 

Then the nominal horse power = — jij — nearly. 

I. Area of Fire G-rate. — The average practice is to give '6^ 
square feet for each nominal horse power. Hence the following 
rule: 

Rule 1. — To find the area of the fire grate. — Multiply the num- 
ber of horses power by -55 ; the product is the area of the fire grate 
in square feet. 

Required the total area of the fire grate for an engine of 400 
horse power. Here total area of fire grate in square feet = 400 X 
'b^ = 220. 

A rule may also be found for expressing the area of the fire grate 
in terms of the length of stroke and the diameter of the cylinder. 
For this purpose we have, 

wi ^^ . '55xd^x ^S , ^ ^^ X ^S ^ ^ 
total area ot nre grate = j^ leet = tt^ leet. 

This formula expressed in words gives the following rule. 

Rule 2. — To find the area of fire grate. — Multiply the cube root 
of the length of stroke in feet by the square of the diameter in in- 
ches ; divide the product by 86 ; the quotient is the area of fire 
grate in square feet. 

Required the total area of the fire grate for an engine whose 
stroke = 8 feet, and diameter of cylinder = 50 inches. 

Here, according to the rule, _ 

502 X ^8 2500 X 2 
total area of fire grate m square feet = w^ = o^ = 

5000 ^^ 

~qw- = 59 nearly. 

In order to work this example by the first rule, we find the 
nominal horse power of the engine whose dimensions we have spe- 
cified is 104*3 ; hence, 

total area of fire grate in square feet = 106*4 x '55 = 58*5. 



256 THE PRACTICAL MODEL CALCULATOR. 

"With regard to these rules we may remark, not only that they 
are founded on practice, and therefore empyrical, but they are only 
applicable to large engines. When an engine is very small, it re- 
quires a much larger area of fire grate in proportion to its size than 
a larger one. This depends upon the necessity of having a certain 
amount of fire grate for the proper combustion of the coal. 

II. Length of Furnace. — The length of the furnace difi'ers con- 
siderably, even in the practice of the same engineer. Indeed, all 
the dimensions of the furnace depend to a certain extent upon the 
peculiarity of its position. From the difficulty of firing long fur- 
naces efficiently, it has been found more beneficial to restrict the 
length of the furnace to about six feet than to employ furnaces of 
greater length. 

III. Height of Furnace above Bars. — This dimension is variable, 
but it is a common practice to make the height about two feet. 

IV. Capacity of Furnace Chamber above Bars. — The average 
per horse power may be taken at 147 feet. Hence the following 
rule : 

EuLE. — To find the capacity of furnace chaniber above bars. — 
Multiply the number of nominal horses power by 1*17 ; the pro- 
duct is the capacity of furnace chambers above bars in cubic feet. 

V. Areas of Flues or Tubes in smallest part. — The average value 
of the area per horse power is 11-2 sq. in. Hence we have the fol- 
lowing rule : 

EuLE. — To find the total area of the flues or tubes in smallest 
part. — Multiply the number of horse power by 11-2 ; the product 
is the total area in square inches of flues or tubes in smallest part. 

Kequired total area of flues or tubes for the boiler of a steam en- 
gine when the horse power = 400. 

For this example we have, according to the rule, 

Total area in square inches = 400 X 11-2 = 4480. 

TVe may also find a very convenient rule expressed in terms of 

the stroke and the diameter of cylinder. Thus, _ 

n . . , 11-2 X C?^ X ^S 
Total area of tubes or flues m square inches = ■ ^ • 

_ d"" X ^S 
- 4 

VI. Effective Heating Surface. — The effective heating surface of 
flue boilers is the whole of furnace surface above bars, the whole 
of tops of flues, half the sides of flues, and none of the bottoms ; 
hence the effective flue surface is about half the total flue surface. 
In tubular boilers, however, the whole of the tube surface is reckoned 
effective surface. 

EFFECTIVE HEATING SURFACE OF FLUE BOILERS. 

Rule 1. — To find the effective heating surface of marine flue 
boilers of large size. — Multiply the number of nominal horse 
power by 5 ; the product is the area of effective heating surface in 
square feet. 



I 



THE STEAM ENGINE. 257 

Required the effective heating surface of an engine of 400 nomi- 
nal horse power. 

In this case, according to the rule, effective heating surface in 
square feet = 400 X 5 = 2000. 

The effective heating surface may be expressed in terms of the 
length of stroke and the diameter of the cylinder. 

Rule 2. — To find tJie total effective heating surface of marine 
flue boilers. — Multiply the square of the diameter of cylinder in 
inches by the cube root of the length of stroke in feet ; divide the 
product by 10 : the quotient expresses the number of square feet 
of effective heating surface. 

Required the amount of effective heating surface for an engine 
whose stroke = 8 ft., and diameter of cylinder = 50 inches. 

Here, according to Rule 2, effective heating surface in square feet 

_ 50" X ^8 _ 2500 X 2 _ 5000 _ 

- To ~ 10 ~ ~io~ - ^^^' 

To solve this example according to the first rule, we have the 
nominal horse power of the engine equal to 106-4. Hence, ac- 
cording to Rule 2, total effective heating surface in square feet = 
106-4 X 4-92 = 5231. 

EFFECTIVE HEATING SURFACE OF TUBULAR BOILERS. 

The effective heating surface of tubular boilers is about equal to 
the total heating surface of flue boilers, or is double the effective 
surface ; but then the total tube surface is reckoned effective sur- 
face. 

It appears that the total heating surface of flue and tubular ma- 
rine boilers is about the same, namely, about 10 square feet per 
horse power. 

VII. Area of Chimney. — Rule 1. — To find the area of chimney, 
— Multiply the number of nominal horse power by 10*23; the pro- 
duct is the area of chimney in square inches. 

Required the area of the chimney for an engine of 400 nominal 
horse power. 

In this example we have, according to the rule, 
area of chimney in square inches = 400 X 10-23 = 4092. 

We may also find a rule for connecting together the area of the 
chimney, the length of the stroke, and the diameter of the cylinder. 

Rule 2. — To find the area of the chimney. — Multiply the square 
of the diameter expressed in inches by the cube root of the stroke 
expressed in feet ; divide the product by the number 5 ; the quo- 
tient expresses the number of square inches in the area of chimney. 

Required the area of the chimney for an engine whose stroke = 
8 feet, and diameter of cylinder = 50 inches. 

We have in this example from the rule, 

502 X ^"8 2500 X 2 
area oi chimney in square inches = ^ = ? =■ 

1000. 

w2 17 



258 THE PRACTICAL MODEL CALCULATOR. 

To work this example according to the first rule, we find, that 
the nominal horse power of this engine is 104*6 : hence, 

area of chimney in square inches = 104*6 X 10*23 = 1070. 

The latter value is greater than the former one by 70 inches. 
This difi'erence arises from our taking too great a divisor in Rule 2. 
Either of the values, however, is near enough for all practical 
purposes. 

VIII. Water in Boiler. — The quantity of water in the boiler 
difi'ers not only for different boilers, but differs even for the same 
boiler at different times. It may be useful, however, to know the 
average quantity of water in the boiler for an engine of a given 
horse power. 

Rule 1. — To determine the average quantity of water in the 
boiler. — Multiply the number of horse power by 5 ; the product 
expresses the cubic feet of water usually in the boiler. 

This rule may be so modified as to make it depend upon the 
stroke and diameter of the cylinder of engine. 

Rule 2. — To determine the cuhic feet of water usually in the 
boiler. — Multiply together the cube root of the stroke in feet, the 
square of the diameter of the cylinder in inches, and the number 5 ; 
divide the continual product by 47 ; the quotient expresses the cu- 
bic feet of water usually in the boiler. 

Required the usual quantity of water in the boilers of an engine 
whose stroke = 8 feet, and diameter of cylinder 50 inches. 

Here we have from the rule, 

. , ., 5 X 502 X ^8" 5 X 2500 x 2 
cubic feet of water in boiler = js; = -^ 

25000 ^_ 
= .» = 532 nearly. 

The engine, with the dimensions we have specified, is of 106*4 
nominal horse power. Hence, according to Rule 1, 

cubic feet of water in boiler = 106*4 x 5 = 532. 

IX. Area of Water Level. — Rule 1. — To find the area of water 
level. — The area of water level contains the same number of square 
feet as there are units in the number expressing the nominal horse 
power of the engine. 

Required the area of water level for an engine of 200 nominal 
horse power. According to the rule, the answer is 200 square 
feet. 

We add a rule for finding the area of water level when the di- 
ameter of cylinder and the length of stroke is given. 

Rule 2. — To find the area of water level. — Slultiply the square 
of the diameter in inches by the cube root of the stroke in feet ; 
divide the product by 47 ; the quotient expresses the number of 
square feet in the area of water level. 

Required the area of the water level for an engine whose stroke 
is 8 feet, and diameter of cylinder 50 inches. 



THE STEAM ENGINE. 259 

In this case, according to the rule, 

area of water level in .quare feet = ^-^ = 106. 

X. Steam Room. — It is obvious that the steam room, like the 
quantity of water, is an extremely variable quantity, differing, not 
only for different boilers, but even in the same boiler at different 
times. It is desirable, however, to know the content of that part 
of the boiler usually filled with steam. 

KuLE 1. — To determine the average quantity of steam room. — 
Multiply the number expressing the nominal horse power by 3 ; 
the product expresses the average number of cubic feet of steam 
room. 

Required the average capacity of steam room for an engine of 
460 nominal horse power. 

According to the rule, 

Average capacity of steam room = 460 X 3 cubic feet = 1380 
cubic feet. 

This rule may be so modified as to apply when the length of 
stroke and diameter of cylinder are given. 

Rule 2. — Multiply the square of the diameter of the cylinder 
in inches by the cube root of the stroke in feet ; divide the product 
by 15 ; the quotient expresses the number of cubic feet of steam 
room. 

Required the average capacity of steam room for an engine whose 
stroke is 8 feet, and diameter of cylinder 5 inches. 

In this case, according to the rule, 

, . , 50^ X ^8 2500 X 2 5000 

Steam room m cubic leet = 



15 15 15 



333i 



3- 

We find that the nominal horse power of this engine is 106-4 ; 
hence, according to Rule 1, 

average steam room in cubic feet = 106-4 x 3 = 320 nearly. 

Before leaving these rules, we would again repeat that they ought 
not to be considered as rules founded upon considerations for giving 
the maximum effect from the combustion of a given amount of fuel ; 
and consequently the engineer ought not to consider them as inva- 
riable, but merely to be followed as far as circumstances will per- 
mit. We give them, indeed, as the medium value of the very va- 
riable practice of several well-known constructors ; consequently, 
although the proportions given by the rules may not be the best 
possible for producing the most useful effect, still the engineer who 
is guided by them is sure not to be very far from the common prac- 
tice of most of our best engineers. It has often been lamented that 
the methods used by different engine makers for estimating the 
nominal powers of their engines have been so various that we can 
form no real estimate of the dimensions of the engine, from its re- 
puted nominal horse power, unless we know its maker ; but the 



260 THE PRACTICAL MODEL CALCULATOR. 

same confusion exists, also, to some extent, in the construction of 
boilers. Indeed, many things may be mentioned, which have 
hitherto operated as a barrier to the practical application of any 
standard of engine power for proportioning the dijQTerent parts of 
the boiler and furnace. The magnitude of furnace and the extent 
of heating surface necessary to produce any required rate of eva- 
poration in the boiler are indeed known, yet each engine-maker 
has his own rule in these matters, and which he seems to think pre- 
ferable to all others, and there are various circumstances influ- 
encing the result which render facts incomparable unless those cir- 
cumstances are the same. Thus the circumstances that govern the 
rate of evaporation, as influenced by different degrees of draught, 
may be regarded as but imperfectly known. And, supposing the 
difficulty of ascertaining this rate of evaporation were surmounted, 
there would still remain some difficulty in ascertaining the amount 
of power absorbed by the condensation of the steam on its passage 
to the cylinder — the imperfect condensation of the same steam after 
it has worked the piston — the friction of the various moving parts 
of the machinery — and, especially, the difference of effect of these 
losses of power in engines constructed on different scales of magni- 
tude. Practice must often vary, to a certain extent, in the con- 
struction of the different parts of the boiler and furnace of an en- 
gine ; for, independently of the difficulty of solving the general 
problem in engineering, the determination of the maximum effect 
with the minimum of means, practice would still require to vary 
according as in any particular case the desired minimum of means 
was that of weight, or bulk, or expense of material. Again, in es- 
timating the proper proportions for a boiler and its appendages, 
reference ought to be made to the distinction between the " power" 
or " effect" of the boiler, and its " duty." This is a distinction to 
be considered also in the engine itself. The power of an engine 
has reference to the time it takes to produce a certain mechanical 
effect without reference to the amount of fuel consumed ; and, on 
the other hand, the duty of an engine has reference to the amount 
of mechanical effect produced by a certain consumption of fuel, and 
is independent of the time it takes to produce that effect. In ex- 
pressing the duty of engines, it would have prevented much need- 
less confusion if the duty of the boiler had been entirely separated 
from that of the engine, as, indeed, they are two very distinct 
things. The duty performed by ordinary land rotative steam en- 
gines is — 

One horse power exerted by 10 lbs. of fuel an hour ; or, 
Quarter of a million of lbs. raised 1 foot high by 1 lb. of coal ; or, 
Twenty millions of lbs. raised one foot by each bushel of coals. 
Though in the best class of rotative engines the consumption is 
not above half of this amount. 

The constant aim of different engine makers is to increase the 
amount of the duty ; that is, to make 10 lbs. of fuel exert a greater 
effect than one horse power ; or, in other words, to make 1 lb. of 



THE STEAM ENGINE. 



261 



coal raise more than a quarter of a million of lbs. one foot tigh. 
To a great extent they have been successful in this. They have 
caused 5 lbs. of coal to exert the force of one horse power, and even 
in some cases as little as 3|- lbs. ; but in these latter cases the 
economy is due chiefly to expansive action. In some of the engines, 
however, working with a consumption of 10 lbs. of coal per nominal 
horse hower per hour, the power really exerted amounts to much 
more than that represented by 33,000 lbs. lifted one foot high in 
the minute for each horse power. Some engines lift 56,000 lbs. 
one foot high in the minute by each horse power, with a consump- 
tion of 10 lbs. of coal per horse power per hour ; and even this 
performance has been somewhat exceeded without a recourse to ex- 
pansive action. In all modern engines the actual performance 
much exceeds the nominal power ; and reference must be had to 
this circumstance in contrasting the duty of different engines. 

MECHANICAL POWER OF STEAM. 

We may here give a table of some of the properties of steam, 
and of its mechanical efi"ects at difi"erent pressures. This table may 
help to solve many problems respecting the mechanical effect of 
steam, usually requiring much laborious calculation. 













Mechanical Effect in Horse Pq-wer of 1 Lb. 


Pressures. 


Tem- 
perature 
iude- 


Weight 


■Velocitj 


OF STEAir. 






Cubic 




■Without Condensation. 


Condensation. 






fS. 


Foot 
Steam. 


Eiit. 


Eipanaion. 


Eipansion. 


Atmo- 
sphere. 


Lbs. per 
Sq. Inch. 





i- 


i 


i 





i 


i 


i 


1-00 


14-70 


212-00 




0-0364 








32-4 


95-2 


170-5 


913 


150-1 


178-6 


194-6 


1-25 


18-38 


2-23-88 


0-0440 


873 


21-5 


10-1 


32-3 


87-4 


95-9 


158-7 


190-6 


209-9 


1-50 


22-05 


234-32 


0-0529 


1135 


36-4 


39-3 


10-8 


30-6 


99-3 


165-2 


199-6 


221-1 


1-75 


25-72 


242-78 


0-0609 


1295 


47-4 


60-8 


42-5 


11-1 


102-0 


170-0 


•206-2 


229-5 


2-00 


29-40 


250-79 


0-0688 


1407 


55-9 


77-5 


67-0 


43-2 


104-3 


174-2 


212-0 


236-5 


2-25 


33-08 


257-90 


0-0766 


1491 


62-8 


90-9 


86-5 


68-8 


106-2 


177-7 


216-7 


242-4 


2-50 


36-75 


263-93 


0-0344 


1556 


68-4 


101-8 


102-4 


89-6 


107-7 


180-5 


220-5 


247-1 


2-75 


40-42 


269-87 


0-0921 


1608 


73-1 


111-0 


115-8 


107-1 


109-3 


183-2 


2-24-2 


251-6 


3-00 


44-10 


275-00 


0-0998 


1652 


71-1 


118-8 


127-1 


121-9 


110-6 


185-4 


2-27-7 


255-2 


3-35 


47-78 


279-86 


0-1073 


1690 


80-7 


125-6 


137-1 


136-7 


111-7 


187-6 


230-0 


258-7 


3-50 


51-45 


284-68 


0-1148 


17-22 


83-8 


131-5 


145-6 


145-8 


112-7 


189-4 


232-4 


261-6 


3"75 


55-12 


288-66 


0-1225 


1750 


86-5 


136-8 


153-2 


155-6 


113-7 


190-1 


234-7 


264-4 


4-00 


5818 


292-91 


0-1298 


1774 


89-0 


141-5 


160 


164-5 


114-6 


192-8 


236-9 


267-0 


4-50 


66-15 


300-27 


0-1445 


1816 


93-2 


149-8 


171-5 


179-4 


116-2 


195-6 


240-5 


271-4 


5-00 


73-50 


307-94 


0-1590 


1850 


96-8 


156-5 


181-6 


192-0 


117-7 


198-3 


244-1 


275-6 


6-00 


88-20 


320-00 


0-1878 


1904 


102-5 


167-2 


196-5 


211-4 


120-2 


202-6 


249-7 


282-2 


7-00 


102-90 


331-56 


0-2159 


1945 


107-0 


1756 


208-4 


226-5 


122-4 


206-4 


254-6 


288-1 


8-00 


117-60 


340-83 


0-2436 


1978 


110-6 


182-4 


217-9 


238-4 


1-24-3 


209 


258-8 


292-1 


900 


132-30 


351-32 


0-2708 


2006 


113-7 


188-2 


225-9 


248-5 


1260 


212 


262-7 


293-6 


10-00 


147-00 


359-60 


0-2977 


2029 


116-3 


193-0 


232-5 


256-7 


127-5 


215 


266-0 


301-4 


12-50 


183-75 


377-42 


0-3642 


2074 


121-5 


202-5 


245-5 


273-0 


130-7 


220 


272-9 


309-5 


15-00 


220-50 


392-90 


0-4288 


2109 


125-7 


210-0 


255-6 


285-4 


133-4 


225 


278-9 


316-4 


17-50 


257-25 


406-40 


0-4924 


2136 


129-0 


216-0 


263-6 


295-2 


135-7 


229 


283-9 


322-2 


20 


294-00 


418-56 


0-5549 


2159 


131-8 


221-0 


270-3 


305-3 


137-8 


•233 


288-3 


327-2 


25 


367-50 


429-34 


0-6775 


2196 


136-3 


229-1 


281-0 


316-2 


141-2 


238 


295-7 


335-8 


30 


441-00 


457-16 


0-7970 


2226 


1400 


235-6 


289-5 


326-4 


144-2 


244 


302-0 


343-1 



It is quite clear that although there is no theoretical limit to the 
benefit derivable from expansion, there must be a limit in practice, 
arising from the friction incidental to the use of very large cylin- 
ders, the magnitude of the deduction due to uncondensed vapour 
when the steam is of a very low pressure, and other circumstances 
which it is needless to relate. It is clear, too, that while the effi- 



262 THE PRACTICAL MODEL CALCULATOR. 

ciency of the steam is increased by expansive action, the efficiency 
of the engine is diminished, unless the pressure of the steam or the 
speed of the piston be increased correspondingly ; and that an en- 
gine of any given size will not exert the same power if made to ope- 
rate expansively without any other alteration that would have been 
realized if the engine had been worked with the full pressure of the 
steam. In the Cornish engines, which work with steam of 40 lbs. 
on the inch, the steam is cut off at one-twelfth of the stroke ; but 
if the steam were cut off at one-twelfth of the stroke in engines em- 
ploying a very low pressure, it would probably be found that there 
would be a loss rather than a gain from carrying the expansion so 
far, as the benefit might be more than neutralized by the friction 
incidental to the use of so large a cylinder as would be necessary 
to accomplish this expansion ; and unless the vacuum Were a very 
good one, there would be but little difference between the pressure 
of the steam at the end of the stroke and the pressure of the va- 
pour in the condenser, so that the urging force might not at that 
point be sufficient to overcome the friction. In practice, therefore, 
in particular cases, expansion may be carried too far, though theo- 
retically the amount of the benefit increases with the amount of the 
expansion. 

We must here introduce a simple practical rule to enable those 
who may not be familiar with mathematical symbols to determine 
the amount of benefit due to any particular measure of expansion. 
When expansion is performed by an expansion valve, it is an easy 
thing to ascertain at what point of the stroke the valve is shut by 
the cam, and where expansion is performed by the slide valve the 
amount of expansion is easily determinable when the lap and stroke 
of the valve are known. 

Rule. — To find the Increase of Efficiency arising from working 
Steam expansively. — Divide the total length of the stroke by the 
distance (which call 1) through which the piston moves before the 
steam is cut off. The hyperbolic logarithm of the whole stroke ex- 
pressed in terms of the part of the stroke performed with the full 
pressure of steam, represents the increase of efficiency due to ex- 
pansion. 

Suppose that the pressure of the steam working an engine is 45 
lbs. on the square inch above the atmosphere, and that the steam 
is cut off at one-fourth of the stroke ; what is the increase of effi- 
ciency due to this measure of expansion ? 

If one-fourth be reckoned as 1, then four-fourths must be taken 
as 4, and the hyperbolic logarithm of 4 will be found to be 1*386, 
which is the increase of efficiency. The total efficiency of the quan- 
tity of steam expended during a stroke, therefore, which without 
expansion would have been 1, becomes 2*386 when expanded into 4 
times its bulk, or, in round numbers, 2*4. 

Let the pressure of the steam be the same as in the last example, 
and let the steam be cut off at half-stroke : what, then, is the in- 
crease of efficiency ? 



THE STEAM ENGINE. 



263 



Here half the stroke is to be reckoned as 1, and the whole stroke 
has therefore to be reckoned as 2. The hyperbolic logarithm of 2 
is -693, which is the increase of efficiency, and the total efficiency 
of the stroke is 1*693, or 1*7. 

We may here give a table to illustrate the mechanical effect of 
steam under varying circumstances. The table shows the me- 



Total 








Total 








pressure 






Mechanical 


pressure 




Volume of 


Mechanical 


in lbs. 


Correspond- 


Volume of Steam 


eflect of 


in lbs. 


Correspond- 


Steam 


effect of 


per 


ing Tem- 


compared with 


Cubic Inch of 


per 


ing Tem- 


compared 


Cubic Inch 


Square 
Inch. 


perature. 


Water. 


Water. 


Square 


perature. 


with Water. 


of Water. 








Inch. 








1 


103 


20-868 


1739 


51 


284 


544 


2312 


2 


126 


10-874 


1812 


52 


286 


534 


2316 


3 


141 


7437 


1859 


53 


287 


625 


2320 


4 


152 


5685 


1895 


54 


288 


516 


2324 


5 


161 


4617 


1924 


55 


289 


608 


2327 


6 


169 


3897 


1948 


56 


290| 


500 


2331 


7 


176 


3376 


1969 


57 


292 


492 


2335 


8 


182 


2983 


1989 


58 


293 


484 


2339 


9 


187 


2674 


2006 


59 


294 


477 


2343 


10 


192 


2426 


2022 


60 


296 


470 


2347 


11 


197 


2221 


2036 


61 


297 


463 


2351 


12 


201 


2050 


2050 


62 


298 


456 


2355 


13 


205 


1904 


2063 


63 


299 


449 


2359 


14 


209 


1778 


2074 


64 


300 


443 


2362 


15 


213 


1669 


2086 


65 


301 


437 


2365 


16 


216 


1573 


2097 


66 


302 


431 


2369 


17 


220 


1488 


2107 


67 


303 


425 


2372 


18 


223 


1411 


2117 


68 


304 


419 


2375 


19 


226 


1343 


2126 


69 


305 


414 


2378 


20 


228 


1281 


2135 


70 


306 


408 


2382 


21 


231 


1225 


2144 


71 


307 


403 


2385 


22 


234 


1174 


2152 


72 


308 


398 


2388 


23 


236 


1127 


2160 


73 


309 


393 


2391 


24 


239 


1084 


2168 


74 


310 


388 


2394 


25 


241 


1044 


2175 


75 


311 


383 


2397 


26 


243 


1007 


2182 


76 


312 


379 


2400 


27 


245 


973 


2189 


77 


313 


374 


2403 


28 


248 


941 


2196 


78 


314 


370 


2405 


29 


250 


911 


2202 


79 


315 


366 


2408 


30 


252 


883 


2209 


80 


316 


362 


2411 


31 


254 


857 


2215 


81 


317 


358 


2414 


32 


255 


833 


2221 


82 


318 


354 


2417 


33 


257 


810 


2226 


83 


318 


350 


2419 


34 


259 


788 


2232 


84 


319 


346 


2422 


35 


261 


767 


2238 


85 


320 


342 


2425 


36 


263 


748 


2243 


86 


321 


339 


2427 


37 


264 


729 


2248 


87 


322 


335 


2430 


38 


266 


712 


2253 


88 


323 


332 


2432 


39 


267 


695 


2259 


89 


323 


328 


2435 


40 


269 


679 


2264 


90 


324 


325 


2438 


41 


271 


664 


2268 


91 


325 


322 


2440 


42 


272 


649 


2273 


92 


326 


319 


2443 


43 


274 


635 


2278 


93 


327 


316 


2445 


44 


275 


622 


2282 


94 


327 


313 


2448 


45 


276 


610 


2287 


95 


328 


310 


2450 


46 


278 


598 


2291 


96 


329 


307 


2453 


47 


279 


586 


2296 


97 


330 


304 


2455 


48 


280 


575 


2300 


98 


330 


301 


2457 


49 


282 


564 


2304 


99 


331 


298 


2460 


60 


283 


554 


2308 


100 


332 


295 


2462 



264 THE PRACTICAL MODEL CALCULATOR. 

chanical effect of the steam generated from a cubic inch of water. 
Our formula gives the effect of a cubic foot of water ; but it can be 
modified to give the effect of the steam of a cubic inch by dividing 
by 1728. In this manner we find, for the mechanical effect of the 
steam of a cubic inch of water, about 3 (459 -{- t) lbs. raised one 
foot high. The table shows that the mechanical effect increases 
with the temperature. The increase is very rapid for temperatures 
below 212° ; but for temperatures above this the increase is less; 
and for the temperatures used in practice we may consider, with- 
out any material error, the mechanical effect as constant. 

INDICATOR. 

An instrument for ascertaining the amount of the pressure of 
steam and the state of the vacuum throughout the stroke of a steam 
engine. Fitzgerald and Neucumn long employed an instrument 
of this kind, the nature of which was for a long time not generally 
known. Boulton and Watt used an instrument acting upon the 
same principle and equally accurate ; but much more portable. In 
peculiarity of construction it is simply a small cylinder truly bored, 
and into which a piston is inserted and loaded by a spring of suit- 
able elasticity to the graduated scale thereon attached. 

The action of an indicator is that of describing, on a piece of 
paper attached, a diagram or figure approximating more or less to 
that of a rectangle, varying of course with the merits or demerits 
of the engine's productive effect. The breadth or height of the 
diagram is the sum of the force of the steam and extent of the va- 
cuum ; the length being the amount of revolution given to the paper 
during the piston's performance of its stroke. 

To render the indicator applicable, it is commonly screwed into 
the cylinder cover, and the motion to the paper obtained by means 
of a sufficient length of small twine attached to one of the radius 
bars ; but such application cannot always be conveniently effected, 
more especially in engines on the marine principle ; hence, other 
parts of such engines, and other means whereby to effect a proper 
degree of motion, must unavoidably be resorted to. In those of 
direct action the crosshead is the only convenient place of attach- 
ment ; but because the length of the engine's stroke is considerably 
more than the movement required for the paper on the indicator, 
it is necessary to introduce a pulley and axle, by which means the 
various movements are qualified to suit each other. 

When the indicator is fixed and the movement for the paper pro- 
perly adjusted, allow the engine to make a few revolutions previous 
to opening the cock ; by which means a horizontal line will be de- 
scribed upon the paper by the pencil attached, and denominated 
the atmospheric line, because it distinguishes between the effect of 
the steam and that of the vacuum. Open the cock, and if the en- 
gine be upon the descending stroke, the steam will instantly raise 
the piston of the indicator, and, by the motion of the paper with the 
pencil pressing thereon, the top side of the diagram will be formed. 



THE STEAM ENGINE. 



265 



At the termination of the stroke and immediately previous to its 
return, the piston of the indicator is pressed down by the surround- 
ing atmosphere, consequently the bottom side of the diagram is de- 
scribed, and by the time the engine is about to make another de- 
scending stroke, the piston of the indicator is where it first started 
from, the diagram being completed ; hence is delineated the mean 
elastic action of the steam above that of the atmospheric line, and 
also the mean extent of the vacuum underneath it. 

But in order to elucidate more ^ t^ 

clearly by example, take the follow- 
ing diagram, taken from a marine 
engine, the steam being cut off after 
the piston had passed through two- 
thirds of its stroke, the graduated 
scale on the indicator, tenths of an 
inch, as shown at each end of the 
diagram annexed. 

Previous to the cock being 
opened, the atmospheric line AB 
was formed, and, when opened, the 
pencil was instantly raised by the 
action of the steam on the piston 
to C, or what is generally termed 
the starting corner; by the move- 
ment of the paper and at the ter- 
mination of the stroke the line CD 
was formed, showing the force of 
the steam and extent of expansion ; 
from D to E show the moments of "" a 

eduction ; from E to F the quality of the vacuum ; and from F to 
A the lead or advance of the valve ; thus every change in the en- 
gine is exhibited, and every deviation from a rectangle, except that 
of expansion and lead of the valve show the extent of proportionate 
defect. Expansion produces apparently a defective diagram, but 
in reality such is not the case, because the diminished power of the 
engine is more than compensated by the saving in steam. Also 
the lead of the valve produces an apparent defect, but a certain 
amount must be given, as being found advantageous to the working 
of the engine, but the steam and eduction corners ought to be as 
square as possible ; any rounding on the steam corner shows a de- 
fect from want of lead ; and rounding on the eduction corner that 
of the passages or apertures being too small. 

Rule. — To compute the power of an Engine from the Indicator 
Diagram. — Divide the diagram in the direction of its length into 
any convenient number of equal parts, through which draw lines 
at right angles to the atmospheric line, add together the lengths of 
all the spaces taken in measurements corresponding with the scale 
on the indicator, divide the sum by the number of spaces, and the 
X 





V 1 r 1 1 II T-i - 1 1 1 1 1 


/l2 


10.8 ^v 


E 


yi& 


3 \ 
^ 12 \ 


/ 6.6 


3 12.8 


' 7.6 


O 12.8 


7.6 


« 12.8 


7.6 


a 12.8 


7.6 


^ 12.8 


7.6 


S 12.8 


7.6 


< 12.6 / 


7.6 


9^8^^^^ 


■TT-T-,', 1 1 


, , , . . I I 1 > 1 I . 1 1 



266 THE PRACTICAL MODEL CALCULATOR. 

quotient is the mean effective pressure on the piston in lbs. per 
square inch. 

Let the result of the preceding diagram be taken as an example. 
Then, the whole sum of vacuum spaces = 1220 -r- 10 = 12*2 lbs. 
mean effect obtained by the vacuum ; and in a similar manner the 
mean effective pressure of steam is found to be 6*28 lbs., hence the 
total effective force = 18*48 lbs. per square inch. And supposing 
2*5 lbs. per square inch be absorbed by friction, What is the actual 
power of the engine, the cylinder's diameter being 32 inches, and 
the velocity of the piston 226 feet per minute ? 

18*48 — 2*5 = 15*98 lbs. per square inch of net available force. 

^, 322 X -7854 X 15*98 x 226 

Then oqaaa = 88 horses power. 

The line under the diagram and parallel to the atmospheric line 
is jfths distant, and represents the perfect vacuum line, the space 
between showing the amount of force with which the uncondensed 
steam or vapour resists the ascent or descent of the piston at every 
part of the stroke. 

As the mean pressure of the atmosphere is 15 lbs. per square 
inch, and the mean specific gravity of mercury 13560, or 2*037 cu- 
bic inches equal 1 lb., it will of course rise in the barometer at- 
tached to the condenser about 2 inches for every lb. effect of va- 
cuum, and as a pure vacuum would be indicated by 30 inches of 
mercury, the distance between the two lines shows whether there 
is or is not any amount of defect, as sometimes there is a consider- 
able difference in extent of vacuum in the cylinder to that in the 
condenser. 

To estimate hy means of an indicator the amount of effective power 
produced hy a steam engine. — Multiply the area of the piston in 
square inches by the average force of the steam in lbs. and by the 
velocity of the piston in feet per minute ; divide the product by 
33,000, and ^J^ths of the quotient equal the effective power. 

Suppose an engine with a cylinder of 37|- inches diameter, a 
stroke of 7 feet, and making 17 revolutions per minute, or 238 feet 
velocity, and the average indicated pressure of the steam 16*73 lbs. 
per square inch ; required the effective power. 

Area = 1104*4687 inches X 16*73 lbs. X 238 feet 
33000 
133*26 X 7 
= Yq ■ = 93*282 horse power. 

To determine the proper velocity for the piston of a steam engine. — 
Multiply the logarithm of the nih. part of the stroke at which the 
steam is cut off by 2*3, and to the product of which add -7. Mul- 
tiply the sum by the distance in feet the piston has travelled when 
the steam is cut off, and 120 times the square root of the product 
equal the proper velocity for the piston in feet per minute. 



26T 



WEIGHT COMBINED WITH MASS, VELOCITY, FORCE, AND 
WORK DONE. 

CALCULATIONS ON THE PRINCIPLE OF VIS VIVA. — MATERIALS EMPLOYED 
IN THE CONSTRUCTION OF MACHINES. — STRENGTH OF MATERIALS, 
THEIR PROPERTIES. — TORSION, DEFLEXION, ELASTICITY, TENACITIES, 
COMPRESSIONS, ETC. — FRICTION OF REST AND OF MOTION, COEFFICIENTS 
OF ALL SORTS OF MOTION. — BANDS. — ROPES. — WHEELS. — HYDRAU- 
LICS. — NEW TABLES FOR THE MOTION AND FRICTION OF WATER. 

WATER-WHEELS. — WINDMILLS, ETC. 

1. Suppose a body resting on a perfectly smooth table, and, when 
in motion, to present no impediment to the body in its course, but 
merely to counteract the force of gravity upon it ; if this body 
weighing 800 lbs. be pressed by the force of 30 lbs. acting hori- 
zontally and continuously, the motion under such circumstances 
will be uniformly accelerated : what is the acceleration ? 

30 
g^ X 32-2 = 1-2075 feet the second. 

2. What force is necessary to move the above-mentioned heavy 
body, with a 23 feet acceleration, under the same circumstances ? 

23 

^^ X 800 = 57-14285 lbs. 

The second of these examples illustrates the principle that the 
force which impels a body with a certain acceleration is equal to 
the weight of the body multiplied by the ratio of its acceleration 
to that of gravity. The first illustrates the reverse, namely, the 
acceleration with which a body is moved forward with a given force, 
is equal to the acceleration of gravity multiplied by the ratio of the 
force to the weight. 

3. A railway car, weighing 1120 lbs., moves with a 5 feet velo- 
city upon horizontal rails, which, let us suppose, offer no impedi- 
ment to the motion, and is constantly pushed by an invariable 
force of 50 lbs. during 20 seconds : with what velocity is it moving 
at the end of the 20th second, or at the beginning of the 21st 
second ? 

50 
5 + 32-2 X Y^ X 20 = 33-75, the velocity. 

4. A carriage, circumstanced as in the last question, weighs 4000 
lbs. ; its initial velocity is 30 feet the second, and its terminal velo- 
city is 70 feet : with which force is the body impelled, supposing it 
to be in motion 20 seconds? 

(70 - 30) X 4000 

32-2x20 =242-17 lbs. 

We have before noticed that the weight (W), divided by 32-2, or 
(^), gives the mass; that is, 



268 THE PRACTICAL MODEL CALCULATOR. 

Weight 
= mass, 

And, force = mass X acceleration. 

5. Suppose a railway carriage, weighing 6440 lbs., moves on a 
horizontal plane offering no impediment, and is uniformly accele- 
rated 4 feet the second, what continuous force is applied ? 

6440 

-oqTo ~ 2^^ ^^s. ,,mass. 

200 X 4 = 800 lbs., the force applied. 

Bj the four succeeding formulas, all questions may be answered 
that may be proposed relative to the rectilinear motions of bodies 
by a constant force. 

For uniformly accelerated motions : 

F 
V = a -{- 32-2^ X t', 

F 
s =- at + 16-1 ^ X t\ 

For uniformly retarded motions : 

F 
V = a — 32-2^ X ty 

F 
s = at — 16-1 X ^ X t^; 

t = the time in seconds, W = the weight in lbs., F = the force in 
lbs., a = the initial velocity, and v = the terminal velocity. 

6. A sleigh, weighing 2000 lbs., going at the rate of 20 feet a 

second, has to overcome by its motion a friction of 30 lbs. : what 

velocity has it after 10 seconds, and what distance has it described ? 

30 
20 — 32-2 X 2Q0Q X 10 = 15-17 feet velocity. 

30 
20 X 10 - 16-1 X 2000 ^ C^^y "" 175-85 feet, distance de- 
scribed. 

7. In order to find the mechanical work which a draught-horse 
performs in drawing a carriage, an instrument called a dynamome- 
ter, or measure of force, is thus used : it is put into communication 
on one side of the carriage, and on the other with the traces of the 
horse, and the force is observed from time to time. Let 126 lbs. 
be the initial force ; after 40 feet is described, let 130 lbs. be the 
force given by the dynamometer ; after 40 feet more is described, 
let 129 lbs. be the force ; after 40 feet more is passed over, let 140 
lbs. be the force ; and let the next two spaces of 40 feet give forces 
of 130 and 120 lbs. respectively. What is the mechanical work done ? 

126 initial force. 
120 terminal force. 

2)246 

123 mean. 



WEIGHT COMBINED WITH MASS. VELOCITY, ETC. 269 

123 + 130 + 129 + 140 + 130 ,„^ , 

5 = 1'^^-^ 

1304 X 40 X 5 = 26080 units of work. 
The following rule, usually given to find the areas of irregular 
figures, may be applied where great accuracy is required. 

Rule. — To the sum of the first and last, or extreme ordinates, 
add four times the sum of the 2d, 4th, 6th, or even ordinates, and 
twice the sum of the 3d, 5th, 7th, &c., or odd ordinates, not includ- 
ing the extreme ones ; the result multiplied by J the ordinates' 
equidistance will be the sum. 
126 
120 

246 sum of first and last. 

246 + 4 X 130 +~2~x 129 + 4 x 140 + 2 x 130 = 1844. 

1844 X 40 

o ■ = 24586-66 units of work or pounds raised one foot 

high. This rule of equidistant ordinates is of great use in the art of 
ship-building. This application we shall introduce in the proper 
place. 

8. How many units of work are necessary to impart to a carriage 
of 3000 lbs. weight, resting on a perfectly smooth railroad, a velo- 
city of 100 feet ? 

(100? 
2\ 32.2 X 3000 = 465838-2 units. 

A unit of work is that labour which is equal to the raising of a 
pound through the space of one foot. A unit of work is done when 
one pound pressure is exerted through a space of one foot, no matter 
in what direction that space may lie. 

Kane Fitzgerald, the first that made steam turn a crank, and 
patented it, and the fly-wheel to regulate its motion, estimated that 
a horse could perform 33000 units of work in a minute, that is, 
raise 33000 lbs. one foot high in a minute. To perform 465838*2 
units of work in 10 minutes would require the application 1*4116 
horse power. 

9. What work is done by a force, acting upon another carriage, 
under the same circumstances, weighing 5000 lbs., which transforms 
the velocity from 30 to 50 feet ? 

(30)2 

644 "" 13-9907, the height due to 30 feet velocity. 

my 

^^ = 38-8043, the height due to 50 feet velocity. 

From 38-8043 
Take 13*9907 



24-8136 

5000 

124068-0000 



x2 



270 THE PRACTICAL MODEL CALCULATOR. 

.-. 124068 are the units of work, and just so mucli work will the 
carriage perform if a resistance be opposed to it, and it be gradu- 
ally brought from a 50 feet velocity to a 30 feet velocity. 

The following is without doubt a very simple formula, but the 
most useful one in mechanics ; by it we have solved the last two 
questions : 

Fs = (H - h) W. 

This simple formula involves the principle technically termed the 
principle of vis viva, or living forces. H is the height due to 

one velocity, say v or H = o" and A, the height due to another a, 
ox h — o-- The weight of the mass = W ; the force F, and the 



To express this principle in words, we may say, that the working 
power (Fs) which a mass either acquires when it passes from a lesser 
velocity {a) to a greater velocity (v), or produces when it is com- 
pelled to pass from a greater velocity {y) into a less (a), is always 
equal to the product of the weight of the mass and the difference 
of the heights due to the velocities. 

When we know the units of tvorh, and the distance in which the 
change of velocity goes on, the force is easily found ; and when the 
force is known, the distance is readily determined. Suppose, in the 
last example, that the change of velocity from 30 to 50 feet took 
place in a distance of 300 feet, then 

124068 
OAA = 413-56 lbs. = F, the force constantly applied during 

300 feet. 

10. If a sleigh, weighing 2000 lbs., after describing a distance of 
250 feet, has completely lost a velocity of 100 feet, what constant 
resistance does the friction offer ? 

Since the terminal velocity = 0, the height due to it = 0, hence 

(100? 2000 
644 x^^ = 1242-2352 lbs. 

We have been calculating upon the principle of vis viva ; but the 
product of the mass and the square of the velocity, without attach- 
ing to it any definite idea, is termed the vis viva, or living force. 

11. A body weighing 2300 lbs. moves with a velocity of 20 feet 
the second, required the vis viva ? 

2300 

-^^ = 71-42857 lbs., mass. 

71-42857 X (20)2 _ 28571-428, the amount of m viva. 
Hence, if a mass enters from a velocity a, into another v, the 
unit of work done is equal to half the difference of the vis viva, at 
the commencement and end of the change of velocity. 
For if the mass be put = M, and W the weight, 



STRENGTH OF MATERIALS. 271 

W • ^r '^^^ 

Then M = — , and the vis viva to velocity a = Ma^ = —-- ; 

if y 

^Nv^ 

and the vis viva to velocity v = Mv^ = . 

r\W Wan /v^ a\ ^ ,„ ,x-„. . 
Then J |— - — I = (^-^) X W^ (H - A) W, for 

v^ a^ 

o— and -^j give the heights due to the velocities v and a, respec- 
tively. The useful formula 

Fs = (H - h) W, 
before given, page 270, may be applied to variable as well as to 
constant forces, if, instead of the constant force F, the mean value 
of the force be applied. 



STRENGTH OF MATERIALS. 

ON MATERIAL EMPLOYED IN THE CONSTRUCTION OP MACHINES. 

In theoretical mechanics, we deal with imaginary quantities, which 
are perfect in all their properties ; they are perfectly hard, and 
perfectly elastic ; devoid of weight in statics and of friction in dy- 
namics. In practical mechanics, we deal with real material objects, 
among which we find none which are perfectly hard, and none, ex- 
cept gaseous bodies, which are perfectly elastic ; all have weight, 
and experience resistance in dynamical action. Practical mechanics 
is the science of automatic labour, and its objects are machines and 
their applications to the transmission, modification, and regulation 
of motive power. In this it takes as a basis the theoretical deduc- 
tions of pure mechanics, but superadds to the formulae of the ma- 
thematician a multitude of facts deduced from observation, and ex- 
perimentally elaborates a new code of laws suited to the varied con- 
ditions to be fulfilled in the economy of the industrial arts. 

In reference to the structure of machines, it is to be observed 
that however simple or complex the machine may be, it is of im- 
portance that its parts combine lightness with strength, and rigidity 
with uniformity of action ; and that it communicates the power 
without shocks and sudden changes of motion, by which the passive 
resistances may be increased and the efi"ect of the engine dimi- 
nished. 

To adjust properly the disposition and arrangement of the indi- 
vidual members of a machine, implies an exact knowledge and esti- 
mate of the amount of strain to which they are respectively subject 
in the working of the machine ; and this skill, when exercised in 
conjunction with an intimate acquaintance with the nature of the 
materials of which the parts are themselves composed, must con- 
tribute to the production of a machine possessing the highest amount 
of capability attainable with the given conditions. 

Materials. — The material most commonly employed in the con- 



272 THE PRACTICAL MODEL CALCULATOR. 

struction of machinery is iron, in the two states of cast and wrought 
or forged iron ; and of these, there are several varieties of quality. 
It becomes therefore a problem of much practical importance to 
determine, at least approximately, the capabilities of the particular 
material employed, to resist permanent alteration in the directions 
in which they are subjected to strain in the reception and trans- 
mission of the motive power. 

To indicate briefly the fundamental conditions which determine 
the capability of a given weight and form of material to resist a 
given force, it must, in the first place, be observed, that rupture 
may take place either by tension or by compression in the direc- 
tion of the length. To the former condition of strain is opposed 
the tenacity of the material ; to the other is opposed the resistance 
to the crushing of its substance. Rupture, by transverse strain, is 
opposed both by the tenacity of the material and its capability to 
withstand compression together of its particles. Lastly, the bar 
may be ruptured by torsion. Mr. Oliver Byrne, the author of the 
present work, in his New Theory of the Strength of Materials has 
pointed out new elements of much importance. 

The capabilities of a material to resist extension and compression 
are often different. Thus, the soft gray variety of cast iron offers 
a greater resistance to a force of extension than the white variety 
in a ratio of nearly eight to five; but the last offers the greatest 
resistance to a compressing force. 

The resistance of cast iron to rupture by extension varies from 
6 to 9 tons upon the square inch ; and that to rupture by compres- 
sion, from 36 to Qd tons. The resistance to extension of the best 
forged iron may be reckoned at 25 tons per inch ; but the corre- 
sponding resistance to compression, although not satisfactorily ascer- 
tained, is generally considered to be greatly less than that of cast 
iron. Roudelet makes it 31J tons on the square inch. Cast iron 
(and even wood) is therefore to be preferred for vertical supports. 

The forces resisting rupture are as the areas of the sections of 
rupture, the material being the same ; this principle holds not only 
in respect of iron, but also of wood. Many inquiries have been in- 
stituted to determine the commonly received principle, that the 
strength of rectangular beams of the same width to resist rupture 
by transverse strain is as the squares of the depths of the beams. 

In these respects the experiments, although valuable on account 
of their extent and the care with which they were conducted, pos- 
sess little novelty ; but in directing attention to the elastic proper- 
ties of the materials experimented upon, it was found that the re- 
ceived doctrine of relation between the limit of elasticity and weight 
requires modification. The common assumption is, that the de- 
struction of the elastic properties of a material, that is, the dis- 
placement beyond the elastic limit, does not manifest itself until 
the load exceeds one-third of the breaking weight. It w\as found, 
however, on the contrary, that its effect was produced and mani- 
fested in a permanent set of the material when the load did not ex- 



STRENGTH OP MATERIALS. 273 

ceed one-sixteentJi of that necessary to produce rupture. Thus a 
bar of one inch square, supported between props 4i feet apart, did 
not break till loaded with 496 lbs. but showed a permanent deflec- 
tion or set when loaded with 16 lbs. In other cases, loads of 7 lbs. 
and 14 lbs. were found to produce permanent sets when the break- 
ing weights were respectively 364 lbs. and 1120 lbs. These sets 
were therefore given by J^^d and ^oth of the breaking weights. 

Since these results were obtained, it has been found th2it time 
and the weight of the material itself are sufficient to effect a per- 
manent deflection in a beam supported between props, so that there 
would seem to be no such limits in respect to transverse strain as 
those known by the name of elastic limits, and consequently the 
principle of loading a beam within the elastic limit has no founda- 
tion in practice. The beam yields continually to the load, but with 
an exceedingly slow progression, until the load approximates to the 
breaking weight, when rupture speedily succeeds to a rapid deflection. 

As respects the effect of tension and compression by transverse 
strain, it was ascertained by a very ingenious experiment that equal 
loads produced equal deflections in both cases. 

Another most important principle developed by experiments, is 
that respecting the compression of supporting columns of different 
heights. When the height of the column exceeded a certain limit, 
it was found that the crushing force became constant, and did not 
increase as the height of the column increased, until it reached 
another limit at which it began to yield, not strictly by crushing, 
but by the bending of the material. The first limit was found to be 
a height of little less than three times the radius of the column ; 
and the second double that height, or about six times the radius of 
the column. In columns of different heights betw^een these limits, 
having equal diameters, the force producing rupture by compression 
was nearly constant. When the column was less than the lower 
limit, the crushing force became greater, and when it was greater 
than the higher limit, the crushing force became less. It was fur- 
ther found that in all cases, where the height of the column was 
exactly above the limits of three times the radius, the section of 
rupture was a plane inclined at nearly the same constant angle of 
55 degrees to the axis of the column. These facts mutually ex- 
plain each other ; for in every height of column above the limit, 
the section of rupture being a plane at the same angle to the axis 
of the column, must of necessity be a plane of the same size, and 
therefore in each case the cohesion of the same number of particles 
must be overcome in producing rupture. And further, the same 
number of particles being to be overcome under the same circum- 
stances for every different height, the same force will be required 
to overcome that amount of cohesion, until at double the height 
(three diameters) the column begins to bend under its load. This 
height being surpassed, it follows that a pressure which becomes 
continually less as the length of the column is increased, will be 
sufficient to break it. 

18 



274 



THE PRACTICAL MODEL CALCULATOR. 



This property, moreover, is not confined to cast iron ; the ex- 
periments of M. Rondelet show that with columns of wrought iron, 
wood, and stone, similar results are obtained. 

From these facts then, it appears that if supporting columns be 
taken of different diameters, and of heights so great as not to allow 
of their bending, yet sufficiently high to allow of a complete sepa- 
ration of the planes of fracture, that is, of heights intermediate to 
three times and six times their radius, then will their strengths be 
as the number of particles in their planes of fracture ; and the 
planes of fracture being inclined at equal angles to the axes of the 
columns, their areas will be as the transverse sections of the co- 
lumns, and consequently the strengths of the columns will be as 
their transverse sections respectively. Taking the mean of three 
experiments upon a column \ inch diameter, the crushing force was 
6426 lbs. ; whilst the mean of four experiments, conducted in ex- 
actly the same manner, upon a column of f of an inch diameter, 
gave 14542 lbs. The diameters of the columns being 2 to 3, the 
areas of transverse section were therefore 4 to 9, which is very 
nearly the ratio of the crushing weights. 

When the length of the column is so great that its fracture is 
produced wholly by bending of its material, the limit has been fixed 
for columns of cast iron, at 30 times the diameter when the ends 
are flat, and 15 times the diameter when the ends are rounded. In 
shorter columns, fracture takes place partly by crushing and partly 
by bending of the material. When the column is enlarged in the 
middle of its length from one and a half to two times the diameter 
of the ends, the strength was found by the same experimenter to be 
greater by one-seventh than in solid columns containing the same 
quantity of iron, in the same length, with their extremities rounded ; 
and stronger by an eighth or a ninth when the extremities were flat 
and rendered immovable by disks. 

The following formulas give the absolute strength of cylindrical 
columns to sustain pressure in the direction of their length. In 
these formulas 

D = the external diameter of the column in inches. 
d = the internal diameter of hollow columns in inches. 
L = the length of the column in feet. 
W = the breaking weight in tons. 



Character of the column. 



Solid cylindrical co 
lumn of cast iron, 

Hollow cylindrical co 
lumn of cast iron, 

Solid cylindrical co- 
lumn of wrought iron 



:} 



Length of the column exceeding 15 
times its diameter. 



Both ends rounded. 

J) 3.76 

W = 14-9 j^ 

D ="•"= — 
W = 13 ^.r. 



W = 42- 



Length of the column exceeding 
times its diameter. 



Both ends flat. 

W= 44-16 j^-T^ 



W = 44-34 
W 



p3-55 

133-75 j^ 



For shorter columns, if W' represent the weight in tons which 
would break the column by bending alone, as given by the preced- 



STRENGTH OF MATERIALS. 275 

ing formulas, and W the weight in tons which would crush the co- 
lumn without bending it, as determined from the subjoined table, 
then the absolute breaking weight of the column W, is represented 
in tons by the formula, 

These rules require the use of logarithms in their applications. 

"When a beam is deflected by transverse strain, the material on 
that side of it on which it sustains the strain is compressed, and the 
material on the opposite side is extended. The imaginary surface 
at which the compression terminates and the extension begins — at 
which there is supposed to be neither extension nor compression — 
is termed the neutral axis of the beam. What constitutes the 
strength of a beam is its resistance to compression on the one side 
and to extension on the other side of that axis — the forces acting 
about the line of axis like antagonist force at the two extremities 
of a lever, so that if either of them yield, the beam will be broken. 
It becomes, however, a question of importance to determine the re- 
lation of these forces ; in other words, to determine whether the 
beam of given form and material will yield first to compression or 
to extension. This point is settled by reference to the columns of 
the subsequent table, page 280, in which it will be observed that the 
metals require a much greater force to crush them than to tear them 
asunder, and that the ivoods require a much smaller force. 

There is also another consideration which must not be overlooked. 
Bearing in mind the condition of antagonism of the forces, it is ob- 
vious, that the further these forces are placed from the neutral 
axis, that is, from the fulcrum of their leverage, the greater must 
be their effect. In other words, all the material resisting compres- 
sion will produce its greatest efiect when collected the farthest possi- 
ble from the neutral axis at the top of the beam ; and, in like man- 
ner, all the material resisting extension will produce its greatest 
effect when similarly disposed at the bottom of the beam. We are 
thus directed to the first general principle of the distribution of the 
material into two flanges — one forming the top and the other the bot- 
tom of the beam — joined by a comparatively slender rib. Associat- 
ing with this principle the relation of the forces of extension and 
compression of the material employed, we arrive at a form of beam 
in which the material is so distributed, that at the instant it is about 
to break by extension on the one side, it is about to break by com- 
pression on the other, and consequently is of the ^^ 
strongest form. Thus, supposing that it is re- | 
quired to determine that form in a girder of i 
cast iron : the ratio of the crushing force of n 
that metal to the force of extension may be 1 



taken generally as GJ to 1, which is therefore also the ratio of the 
lower to the upper flange, as in the annexed sectional diagram. 

A series of nine castings were made, gradually increasing the 
lower flange at the expense of the upper one, and in the first eight 



276 



THE PRACTICAL MODEL CALCULATOR. 



experiments tlie beam broke by tbe tearing asunder of the lower 
flange ; and in the last experiment the beam yielded by the crush- 
ing of the upper flange. In the eight experiments the upper flange 
was therefore the weakest, and in the ninth the strongest, so that 
the form of maximum strength was intermediate, and very closely 
allied to that form of beam employed in the last experiment, which 
was greatly the strongest. The circumstances of these experiments 
are contained in the following table. 



No. of experi- 


Eatio of surfaces of com- 


Area of cross sections 


strength per sq. inch 


ments. 


pression and extension. 


in sq. inches. 


of sections in lbs. 


1 


Itol 


2-82 


2368 


2 


lto2 


2-8T 


256T 


3 


lto4 


3-02 


273T 


4 


lto4i 


3-3T 


3183 


5 


lto4 


4-50 


3214 


6 


1 to 5J 


5-00 


3346 


T 


1 to 3i 


4-628 


3246 


8 


1 to 4-3 


5-86 


331T 


9 


1 to 6-1 


64 


4075 



To determine the weight necessary to hrealc beams cast according 
to the form described : 

Multiply the area of the section of the lower flange by the depth 
of the beam, and divide the product by the distance bdlween the 
two points on which the beam is supported : this quotient multi- 
plied by 536 when the beams are cast erect, and by 514 when they 
are cast horizontally, will give the breaking weight in cwts. 

From this it is not to be inferred that the beam ought to have 
the same transverse section throughout its length. On the con- 
trary, it is clear that the section ought to have a definite relation 
to the leverage at which the load acts. From a mathematical con- 
sideration of the conditions, 
it indeed appears that the 
efi'ect of a given load to 
break the beam varies when 
it is placed over diff'erent 
points of it, as the products 
of the distances of these points from the points of support of the 
beam. Thus the efi'ect of a weight pkced at the point W^ is to the 
efi'ect of the same weight acting upon the point W^, as the product 
AW, X W, B is to the product AW^ X W^ B ; the points of sup- 
port being at A and B. Since then the efi'ect of a weight increases 
as it approaches the middle of the length of the beam, at which it 
is a maximum, it is plain that the beam does not require to have 
the same transverse section near to its extremities as in the middle ; 
and, guided by the principle stated, it is easy to perceive that its 
strength at different points should in strictness vary as the products 
of the distances of these points from the points of support. By 



2Y 



Z^ 



STRENGTH OF MATERIALS. 27Y 

taking this law as a fundamental condition in the distribution of 
the strength of a beam, whose load we may conceive to be accumu- 
lated at the middle of its length, we arrive at the strongest form 
which can be attained under given circumstances, with a given 
amount of material ; we arrive at that form which renders the beam 
equally liable to rupture at every point. Now this form of maxi- 
mum strength may be attained in two ways ; either by varying the 
depth of the beam according to the law stated, or by preserving 
the depth everywhere the same, and varying the dimensions of the 
upper and lower flanges according to the same law. The conditions 
are manifestly identical. We may therefore assume generally the 
condition that the section is rectangular, and that the thickness of 
the flanges is constant ; then the outline determined by the law in 
question, in the one case of the elevation of the beam and in the 
other of the plan of the flanges, is the geometrical curve called a 
parabola — rather, two parabolas joined base to base at the middle 
between the points of support. The annexed diagram represents 
the plan of a cast-iron girder according to this form, the depth 




being uniform throughout. Both flanges are of the same form, 
but the dimensions of the upper one are such as to give it only a 
sixth of the strength of the other. 

This, it will be observed, is also the form, considered as an ele- 
vation, of the beam of a steam engine, which good taste and regard 
to economy of material have rendered common. 

It must, however, be borne in mind, that in the actual practice 
of construction, materials cannot with safety be subjected to forces 
approaching to those which produce rupture. In machinery espe- 
cially, they are liable to various and accidental pressures, besides 
those of a permanent kind, for which allowance must be made. 
The engineer must therefore in his practice depend much on expe- 
•rience and consideration of the species of work which the engine is 
designed to perform. If the engine be intended for spinning, 
pumping, blowing, or other regular work, the material may be sub- 
jected to pressures approaching two-thirds of that which would ac- 
tually produce rupture ; but in engines employed to drive bone- 
mills, stampers, breaking-down rollers, and the like, double that 
strength will often be found insufficient. In cases of that nature, 
experience is a better guide than theory. 

It is also to be remarked that we are often obliged to depart 
from the form of strength which the calculation gives, on account 
of the partial strains which would be put upon some of the parts 
of a casting, in consequence of unequal cooling of the metal when 
the thicknesses are unequal. An expert founder can often reduce 
the irregular contractions which thus result ; but, even under the 
best management, fracture is not unfrequently produced by irregu- 



278 THE PRACTICAL MODEL CALCULATOR. 

larity of cooling, and it is at all times better to avoid tlie danger 
entirely, than to endeavour to obviate it by artifice. For this rea- 
son, the parts of a casting ought to be as nearly as possible of such 
thickness as to cool and contract regularly, and by that means all 
partial strain of the parts will be avoided. 

With respect to design, it is also to be remarked, that mere theo- 
retical properties of parts will not, under all the varieties of circum- 
stances which arise in the working of a machine, insure that exact 
adjustment of material and propriety of form so much desired in 
constructive mechanics. Every design ought to take for its basis 
the mathematical conditions involved, and it would, perhaps, be im- 
possible to arrive at the best forms and proportions by any more 
direct mode of calculation ; but it is necessary to superadd to the 
mathematical demonstration, the exercise of a well-matured judg- 
ment, to secure that degree of adjustment and arrangement of parts 
in which the merits of a good design mainly consist. A purely 
theoretical engine would look strangely deficient to the practised 
eye of the engineer ; and the merely theoretical contriver would 
speedily find himself lost, should he venture beyond his construction 
on paper. His nice calculations of the " work to be performed," of 
the vis viva of the mechanical organs of his machine, and of the modu- 
lus of elasticity of his material, would, in practice, alike deceive him. 

The first consideration in the design of a machine is the quantity 
of work which each part has to perform — in other words, the forces, 
active and inactive, which it has to resist ; the direction of the 
forces in relation to the cross-section and points of support ; the 
velocity, and the changes of velocity to which the moving parts are 
subject. The calculations necessary to obtain these must not be 
confined to theory alone ; neither should they be entirely deduced 
by "rule of thumb ;" by the first mode the strength would, in all 
probability, be deficient from deficiency of material, and by the 
second rule the material would be injudiciously disposed ; weight 
would be added often where least needed, merely from the deter- 
mination to avoid fracture, and in consequence of a want of know- 
ledge respecting the true forms best adapted to give strength. 

To the following general principles, in practice, there are but 
few real exceptions : 

I. Direct Strain. — To this a straight line must be opposed, and 
if the part be of considerable 
length, vibration ought to be coun- 
teracted by intersection of planes, 
(technically feathers,) as repre- 
sented in the annexed diagrams, 
or some such form, consistent with the purpose for which the part 
is intended. 

II. Transverse Strain. — To this a parabolic form of section must 
be opposed, or some simple figure including the parabolic form. 
For economy of material, the vertex of the curve ought to be at 
the point where the force is applied; and when the strain passes 




STRENGTH OP MATERIALS. 279 

alternately from one side of the part to the other, the curve ought 
to be on both sides, as in the beam of a steam engine. 

When a loaded piece is supported at one end only, if the breadth 
foe everywhere the same, the form of equal strength is a triangle ; 
but, if the section be a circle, then the solid will be that generated 
by the revolution of a semi-parabola about its longer axis. In prac- 
tice, it will, however, be sufficient to employ the frustum of a cone, 
of which, in the case of cast iron, the diameter at the unsupported 
end is one-third of the diameter at the fixed end. 

III. Torsion. — The section most commonly opposed to torsion 
is a circle ; and, if the strain be applied to a cylinder, it is obvious 
the rupture must first take place at the surface, where the torsion is 
greatest, and that the further the material is placed from the neutral 
axis, the greater must be its power of resistance ; and hence, the 
amount of materials being the same, a shaft is stronger when made 
hollow than if it were made solid. 

It ought not, however, to be supposed that the circle is the only 
figure which gives an axis the property of offering, in every direc- 
tion, the same resistance to flexure. On the contrary, a square sec- 
tion gives the same resistance in the direction of its sides, and of 
its diagonals ; and, indeed, in every direction the resistance is equal. 
This is, moreover, the case with a great number of other figures, 
which may be formed by combining the circle and the square in a 
symmetrical manner ; and hence, if the axis, strengthened by salient 
sides, as in feathered shafts, do not answer as well as cylindrical 
ones, it must arise from their not being so well disposed to resist 
torsion, and not from any irregularities of flexure about the axis 
inherent in the particular form of section. 

This subject has been investigated with much care, and, accord- 
ing to M. Cauchy, the modulus of rupture by torsion, T, is con- 
nected with the modulus of rupture by transverse strain S, by the 
simple analogy T = | S. 

The forms of all the parts of a machine, in whatever situation 
and under every variety of circumstances, may be deduced from 
these simple figures ; and, if the calculations of their dimensions 
be correctly determined, the parts Avill not only possess the requi- 
site degree of strength, but they will also accord with the general 
principles of good taste. 

In arranging the details of a machine, two circumstances ought 
to be taken into consideration. The first is, that the parts subject 
to wear and influenced by strain, should be capable of adjustment ; 
the second is, that every part should, in relation to the work it has 
to perform, be equally strong, and present to the eye a figure that 
is consistent with its degree of action. Theory, practice, and taste 
must all combine to produce such a combination. No formal law 
can be expressed, either by words or figures, by which a certain 
contour should be preferred to another ; both may be equally strong 
and equally correct in reference to theory ; custom, then, must be 
appealed to as the guide. 



280 



THE PRACTICAL MODEL CALCULATOR. 



TABLES OF THE MECHANICAL PROPERTIES OF THE MATERIALS MOST 
COMMONLY EMPLOYED IN THE CONSTRUCTION OF MACHINES AND 
FRAMINGS. 



NAMES. 


SDecifio 
Glavitj. 


■Weight of 

leucjoft. 

in lbs. 


Tenacity per 

sq.uare inch 

in lbs. 


Crushing 


Modulus of 

elasticity 

in lbs. 


Mod. of 


Crushing 
force to 
tenacity. 


Table I.— Mechanical Properties of 
tJie Cotmwn Metals. 

Brass (cast) 

Copper (cast) 


8-399 
8-607 


525-00 
537-93 
54906 
660-00 


17968 
19072 

61228 

48000 
253^ tons 

253^ tons 

30 tons 

27 tons 

32 tons 
36 to 43 tons 
60 to 91 tons 

14 tons 
18 tons 

213^ tons 

25 tons 

13.505 
16683 
14200 
17755 

21907 
17466 
13434 
18855 
16676 

1824 
3328 
2581 
40902 

120000 

150000 

5322 

16000 
15784 
17850 
15000 

12400 

17600 
7000 
13489 

12000 
10220 
11800 
16500 

17300 

10253 

12780 
7818 

11700 

7200 

15000 
14000 
8000 


10304 

108540 
106375 
115442 
133440 

145435 
93366 
86397 
81770 
82739 

7733 f 
9363 t 
6402 
11633 

10331 

6000 
5568 

8198 
4684 wet -1 
9504 dry/ 
4231 wet) 
9509 dry/ 

5375 
5445 

3107 wet) 
5124 dry/ 

12101 


8930000 


48240 
45360 
41046 
45828 
37503 
38556 

42120 
36288 
43497 
37503 
35316 
33104 
33145 
43541 
40159 
2^552 
&5923 
33850 
34862 

11202 
93363 
10920 
9624 
9864 
10386 

6078 
6612 

"6894 

10032 

10596 

8748 
9792 
8946 

14772 


0-573:1 

8037 : 1 
6-376 : 1 
8-129:1 
7-515 : 1 

5-346 : 1 
6-431 : 1 

4-337 : 1 
4-961:1 




ditto (wire-drawn) .... 

ditto (in bolts) 

Iron (English wrought) .... 

ditto (in bars) 

ditto (hammered) 

ditto (Russian) in bars .... 
ditto (Swedish) in bars .... 
ditto (English) in wire, 10th inch diam. 
ditto (Russian) in wire, l-20th to l-30th 
inch diameter .... 
ditto (rolled in sheets and cut lengthwise) 
ditto cut crosswise .... 
ditto in chains, oval links, 6 inches clear, 

iron }^ inch diameter . 
ditto (Brunton's) with stay cross link . 
Cast-iron (Old Park) 

ditto (Adelphi) .... 

ditto (Alfreton) 

ditto (scrap) 

ditto (Carron, No. 2) hot blast . 

ditto ( do. do. ) cold blast . 

ditto f do. No. 3) . 

ditto ( do. do. ) hot blast . 

ditto (Devon, No. 3) cold blast 

ditto ( do. do. ) hot blast . 

ditto fBuffirey.No, 1) cold blast . . 

ditto ( do. do. ) hot blast 

ditto f Coed-Talon, No. 2) cold blast 

ditto ( do. do. ) hot blast . 

ditto ( do. No. 3) cold blast 

ditto ( do. do. ) hot blast . 

ditto (Milton, No. 1) hot blast . . 

ditto (Muirkirk, No. 1) cold bla^t . 

ditto ( do. do. ) hot blast . 

ditto (Elsicar, No. 1) cold blase . . 
Lead (English cast) .... 

ditto (milled-sheet) 

ditto (wire-drawn 

Silver (standard) 

Mercury (at 320) 

ditto (at60O) 

Steel rsoft) 

ditto (razor-tempered) .... 

Tin (cast) 

Zinc (cast) 

ditto (rolled 

Table 11.— Principal Woods. 
Acacia (English) 

i^-«^ {iv; • . • . • • • • 

Bi-MimTi^cL- . •.'.•.•. 

fChristiania middle . 
P, , 1 Memel middle .... 
"" ) Norway spruce 

(English 

Elm (seasoned) 

«' {£/. '?"*■'* .•.-.•.■ 

Larch (seasoned) 

Lignum-vita 

Mahogany (Spanish) . . ... 

[-English 


8-878 

7-700 
f 7-760 
17-800 

}E 
}~ 

7^ 
7-066 
7-094 
7-056 
7-295 
7-229 
7-079 
6-998 
6-955 
6-963 
7-104 
6-970 
6-976 
7-113 
6-953 
7-030 
11-446 
11-407 
11-317 
10-312 
13-619 
13-580 
7-780 
7-840 
7-291 
7-028 
7-215 

0-71 
0-854 
0-690 
0-792 
0648 
0-698 
0-590 
0-340 
0-470 
0-588 
0-553 
0-753 
0-522 
1-220 
0-800 

0-934 






481-20 
475-50 
487-00 


24920000 






































18014400 
18353600 
17686400 
18032000 
16085000 
17270500 
16246966 
17873100 
23907700 
224736.50 
15381200 
13730500 
14313500 
14322500 
17102000 
14707900 
11974500 
14003550 
13294400 
13981000 
720000 








440-37 
441-62 
443-37 
441-00 
455-93 
451-81 
442-43 
437-37 
43406 
435-50 
449-62 
435-62 
436-00 
444-56 
434-56 
439-37 
717-45 
712-93 
705-12 
644-50 
851-18 
848-75 
486-25 
490-00 
455-68 
439-25 
450-9 

44-37 
53-37 
43-12 
49-50 
40-50 
43-62 
36-87 
21-25 
29-37 
36-75 
34-56 
47-06 
32-62 
76-25 
50-00 

58-37 

54-50 

47-24 
41-25 
41-06 
28-81 
40-00 

23-93 

4106 
24-37 
50-43 
















i 

! 






29000000 
4608000 
13680000 


1 




1152000 
13536000 
1562400 
1257600 
1672000 
1535200 


0-55:1 
0-43:1 






1 






699840 
2191200 


0-79-1 


0-50:1 
0-55:1 


1052800 




0-50 : 1 
/0-28:l 
i 0-57:1 
/0-42:l 
1 0-95:1 


1451200 

2148800 

1191200 
1225600 
1840000 
1600000 

2414400 


(Dantzic 

rPitch 

Pine ^Red 

(Yellow 

Plane-tree 

Poplar 

Teak (dry) 

Willow (dry) 

Yew (Spanish) 


0-756 
0-660 
0-6.57 
0-461 
0-64 

0-383 

0-6.57 
0-390 
0-807 








/0-43:l 
i0-74:l 

0-81:1 









STRENGTH OP MATERIALS. 



281 



THE COHESIVE STRENGTH OF BODIES. 

The following Table contains the result of experiments on the 
cohesive strength of various bodies in avoirdupois pounds ; also, 
one-third of the ultimate strength of each body, this being consi- 
dered sufficient, in most cases, for a permanent load : 



Karnes of Bodies. 


Square Bar. 


One-third. 


Bound Bar. 


One-third. 


WOODS. 

Boxwood 


Ibx. 
20000 

17000 
15000 
12000 
11500 
11000 

18656 

55872 

72064 

133152 

124400 

134256 

19072 

33792 

17968 

4736 

1824 


lbs. 

6667 
5667 
5000 
4000 
3866 
3667 

6219 
18624 
24021 
44384 
41366 
44752 

6357 
11264 

5989 

1579 
608 


lbs. 

15708 

13357 

11781 

9424 

9032 

8639 

14652 
43881 
56599 

104577 
97703 

105454 

14979 

26540 

14112 

3719 

1432 


lbs. 

5236 
4452 
3927 
3141 
3011 
2880 

4884 

14627 

18866 

34859 

32568 

35151 

4993 

8827 

4704 

1239 

477 


Ash 


Teak 


Fir 


Beach 


Oak 


METALS. 


English wrought iron 

Swedish do. do 




Cast do 








Cast tin 


Cast lead 





PROBLEM I. 

Rule. — To find the ultimate cohesive strength of square, round, 
and rectangular bars, of any of the various bodies, as specified in 
the table. — Multiply the strength of an inch bar, (as in the table,) 
of the body required, by the cross sectional area of square and 
rectangular bars, or by the square of the diameter of round bars ; 
and the product will be the ultimate cohesive strength. 

A bar of cast iron being IJ inches square, required its cohesive 



power. 



1-5 X 1-5 X 18656 = 41976 lbs. 



Required the cohesive force of a bar of English wrought iron, 
2 inches broad, and f of an inch in thickness. 

2 X -375 X 55872 = 41901 lbs. 

Required the ultimate cohesive strength of a round bar of 
wrought copper | of an inch in diameter. 

•75^ X 26540 = 14928-75 lbs. 

PROBLEM n. 

Rule. — The weight of a body being given, to find the cross sec- 
tional dimensions of a bar or rod capable of sustaining that iveight. — 
For square and round bars, divide the weight given by one-third 
of the cohesive strength of an inch bar, (as specified in the table,) 
and the square root of the quotient will be the side of the square, 
or diameter of the bar in inches. 
y2 



282 



THE PRACTICAL MODEL CALCULATOR. 



And if rectangular, divide the quotient by tlie breadth, and the 
result will be the thickness. 

What must be the side of a square bar of Swedish iron to sus- 
tain a permanent weight of 18000 lbs ? 

18000 ^, -, . ^ . T 

■^9Zo9T ^ ' ^^ nearly | of an inch square. 

Required the diameter of a round rod of cast copper to carry 
a weight of 6800 lbs. 

6800 ,,^. , ,. 
\/ .qQo = 1.16 inches diameter. 

A bar of English wrought iron is to be applied to carry a weight 
of 2760 lbs. ; required the thickness, the breadth being two inches. 

2760 

^anc^A = '142 -7- 2 = -071 of an inch in thickness. 
18624 

A Table showing the circumference of a rope equal to a chain 
made of iron of a given diameter, and the weight in tons that 
each is proved to carry ; also, the weight of a foot of chain made 
from iron of that dimension. 



Ropes. 


Chains. 


Proved to carry 


Weight of a lineal 


Cir. in Ins. 


Diam. in Inches. 


in tons. 


foot in lbs. Avr. 


3 


landJg 


1 


1-08 


4 


f 


2 


1-5 


4f 


fandi 


3 


2 


H 


i 


4 


2-7 


6 


landJg 


5 


3-3 


6* 


t 


6 


4 


7 


fandJg 


8 


4-6 


n 


1 


n 


5-5 


8 


fand^ 


Hi 


6-1 


9 


7 
8 


13 


7-2 


9* 


i and Jg 


15 


8-4 


lOJ 


1 inch. 


18 


9-4 



ON THE TRANSVERSE STRENGTH OF BODIES. 



The tranverse strength of a body is that power which it exerts 
in opposing any force acting in a perpendicular direction to its 
length, as in the case of beams, levers, &c., for the fundamental 
principles of which observe the following : — 

That the transverse strength of beams, &c. is inversely as their 
lengths, and directly as their breadths, and square of their depths, 
and, if cylindrical, as the cubes of their diameters ; that is, if a 
beam 6 feet long, 2 inches broad, and 4 inches deep, can carry 
2000 lbs., another beam of the same material, 12 feet long, 2 inches 
broad, and 4 inches deep, will only carry 1000, being inversely as 
their lengths. Again, if a beam 6 feet long, 2 inches broad, and 
4 inches deep, can support a weight of 2000 lbs., another beam of 



STRENGTH OF MATERIALS. 



283 



the same material, 6 feet long, 4 inches broad, and 4 inches deep, 
will support double that weight, being directly as their breadths ; 
— but a beam of that material, 6 feet long, 2 inches broad, and 
8 inches deep, will sustain a weight of 8000 lbs. ; being as the 
square of their depths. 

From a mean of experiments made, to ascertain the transverse 
strength of various bodies, it appears that the ultimate strength 
of an inch square, and an inch round bar of each, 1 foot long, 
loaded in the middle, and lying loose at both ends, is nearly as 
follows, in lbs. avoirdupois. 



Names of Bodies. 


Square Bar. 


One-third. 


Round Bar. 


One-third. 


Oak 


800 

1137 

509 

916 

666 

2580 

4013 


267 

379 
139 
305 
188 
860 
1338 


628 
893 
447 
719 
444 
2026 
3152 


209 
298 
149 
239 
148 
675 
1050 


Ash 


Elm 


Pitch pine 


Deal. 


Wrought iron 



PROBLEM I. 

Rule. — To find the ultimate transverse strength of any rectan- 
gular beam, supported at both ends, and loaded in the middle; or 
supported in the middle, and loaded at both ends ; also, when the 
weight is between the middle and the end ; likeivise when fixed at 
one end and loaded at the other. — Multiply the strength of an inch 
square bar, 1 foot long, (as in the table,) by the breadth, and square 
of the depth in inches, and divide the product by the length in 
feet ; the quotient will be the weight in lbs. avoirdupois. 

What weight will break a beam of oak 4 inches broad, 8 inches 
deep, and 20 feet between the supports ? 
800 X 4 X 8^ 
20 = ^^^^^ ^^^• 

When a beam is supported in the middle, and loaded at each 
end, it will bear the same weight as when supported at both ends 
and loaded in the middle ; that is, each end will bear half the 
weight. 

When the weight is not situated in the middle of the beam, but 
placed somewhere between the middle and the end, multiply twice 
the length of the long end by twice the length of the short end, and 
divide the product by the whole length of the beam ; the quotient 
will be the effectual length. 

Required the ultimate transverse strength of a pitch pine plank 
24 feet long, 3 inches broad, 7 inches deep, and the weight placed 
8 feet from one end. 



32 X 16 



24 
and 



21*3 effective length. 



916 X 3 X 7" 
21^3 



= 6321 lbs. 



284 THE PEACTICAL MODEL CALCULATOR. 

Again, when a beam is fixed at one end and loaded at the other, 
it will only bear ^ of the weight as when supported at both ends 
and loaded in the middle. 

What is the weight requisite to break a deal beam 6 inches broad, 
9 inches deep, and projecting 12 feet from the wall ? 

566 X 6 X 9^ 

j2 = 22^2^ -^ 4 = 5730-7 lbs. 

The same rules apply as well to beams of a cylindrical form, 
with this exception, that the strength of a round bar (as in the 
table) is multiplied by the cube of the diameter, in place of the 
breadth, and square of the depth. 

Required the ultimate transverse strength of a solid cylinder of 
cast iron 12 feet long and 5 inches diameter. 

2026 X 52 

j2 = ^11^^ l^s- 

What is the ultimate transverse strength of a hollow shaft of 
cast iron 12 feet long, 8 inches diameter outside, and containing 
the same cross sectional area as a solid cylinder 5 inches diameter ? 



^8^ - 52 = 6-24, and 8^ - 6-242 ^ 269. 

2026 X 269 
Then, j^ = ^^^^^ ^^s. 

When a beam is fixed at both ends, and loaded in the middle, it 
will bear one-half more than it will when loose at both ends. 

And if a beam is loose at both ends, and the weight laid uni- 
formly along its length, it will bear double; but if fixed at both 
ends, and the weight laid uniformly along its length, it will bear 
triple the weight. 

PROBLEM II. 

Rule. — To find the hreadtli or depth of beams intended to sup- 
j)ort a permanent w eight. —^Ivliv^lj the length between the sup- 
ports, in feet, by the weight to be supported in lbs., and divide the 
product by one-third of the ultimate strength of an inch bar, (as 
in the table,) multiplied by the square of the depth ; the quotient 
will be the breadth, or, multiplied by the breadth, the quotient will 
be the square of the depth, both in inches. 

Required the breadth of a cast iron beam 16 feet long, 7 inches 
deep, and to support a weight of 4 tons in the middle. 

8960 X 16 
4 tons = 8960 lbs. and ^^^ ^^ = 3-4 inches. 

What must be the depth of a cast iron beam 3-4 inches broad, 
16 feet long, and to bear a permanent weight of four tons in the 
middle ? 

8960x16 
^ 860 X 3-4 == ^ '^^^^^- 



STRENGTH OF MATERIALS. 285 

"When a beam is fixed at both ends, the divisor must be multi- 
plied by 1*5, on account of it being capable of bearing one-half 
more. 

When a beam is loaded uniformly throughout, and loose at both 
ends, the divisor must be multiplied by 2, because it will bear 
double the weight. 

If a beam is fast at both ends, and loaded uniformly throughout, 
the divisor must be multipled by 3, on account that it will bear 
triple the weight. 

Required the breadth of an oak beam 20 feet long, 12 inches 
deep, made fast at both ends, and to be capable of supporting a 
weight of 12 tons in the middle. 

26880 X 20 
12 tons = 26880 lbs., and o(^f; y 102 ^ -1.;; = 9-7 inches. 

Again, when a beam is fixed at one end, and loaded at the other, 
the divisor must be multiplied by '25 ; because it will only bear 
one-fourth of the weight. 

Required the depth of a beam of ash 6 inches broad, 9 feet 
projecting from the wall, and to carry a weight of 47 cwt. 

5264 X 9 
47 cwt. = 5264 lbs., and x^ ona x 6 x -25 ~ ^"^^ inches deep. 

And when the weight is not placed in the middle of a beam, the 
efi'ective length must be found as in Problem I. 

Required the depth of a deal beam 20 feet long, and to support 
a weight of 63 cwt. 6 feet from one end. 

28 X 12 

— OQ — = 16-8 effective length of beam, and 

63 cwt. = 7056 lbs. : hence 



7056 X16-8 ^^^^ . ^ , 
-v/^r^Q— -^ — ~ 10-24 inches deep. 

Beams or shafts exposed to lateral pressure are subject to all the 
foregoing rules, but in the case of water-wheel shafts, &c., some al- 
lowances must be made for wear ; then the divisor may be changed 
from 675 to 600 for cast iron. 

Required the diameter of bearings for a water-wheel shaft 12 

feet long, to carry a weight of 10 tons in the middle. 

10 tons = 22400 lbs., and 

22400 

^QQ — -^448 = 7*65 inches diameter. 

And when the weight is equally distributed along its length, the 
cube root of half the quotient will be the diameter, thus : 

448 

—^ = A^224 = 6*07 inches diameter. 

Required the diameter of a solid cylinder of cast iron, for the 
shaft of a crane, to be capable of sustaining a weight of 10 tons ; 



286 



THE PRACTICAL MODEL CALCULATOR. 



one end of tlie shaft to be made fast in the ground, the other to 
project 6J feet ; and the effective leverage of the jib as If to 1. 
10 tons = 22400 lbs., and 
22400 X 6'b X 1-75 

_^5x-25 =1^^^ 

And ^1509 = 11*47 inches diameter. 
The strength of cast iron to wrought iron, in this direction, is as 
9 is to 14 nearly ; hence, if wrought iron is taken in place of cast 
iron in the last example, what must be its diameter ? 



^- 



15 09 X 9 
14 



= 9*89 inches diameter. 



ON TORSION OR TWISTING. 



The strength of bodies to resist torsion, or wrenching asunder, 
is directly as the cubes of their diameters ; or, if square, as the 
cube of one side ; and inversely as the force applied multiplied into 
the length of the lever. 

Hence the rule. — 1. Multiply the strength of an inch bar, by 
experiment, (as in the following table,) by the cube of the diameter, 
or of one side in inches ; and divide by the radius of the wheel, or 
length of the lever also in inches ; and the quotient will be the ul- 
timate strength of the shaft or bar, in lbs. avoirdupois. 

2. — Multiply the force applied in pounds by the length of the 
lever in inches, and divide the product by one-third of the ultimate 
strength of an inch bar, (as in the table,) and the cube root of the 
quotient will be the diameter, or side of a square bar in inches ; 
that is, capable of resisting that force permanently. 

The following Table contains the result of experiments on inch bars, 

of various metals, in lbs. avoirdupois. 



Names of Bodies. 


Kotind Bar. 


One-tMrd. 


Square Bar. 


One-third. 


Cast iron. 


11943 

12063 

11400 

20025 

20508 

21111 

5549 

4825 

1688 

1206 


3981 

4021 
3800 
6675 
6836 
7037 
1850 
1608 
563 
402 


15206 

15360 

14592 

25497 

26112 

26880 

7065 

6144 

2150 

1536 


5069 

5120 
4864 
8499 
8704 
8960 
2355 
2048 
717 
512 


English wrought iron 
Swedish do. do. 
Blistered steel 


Shear do 

Cast do 

Yellow brass 


Cast copper 


Tin 


Lead 





What weight, applied on the end of a 5 feet lever, will wrench 
asunder a 3 inch round bar of cast iron ? 
11943 X 3^ 
go = 5374 lbs. avoirdupois. 

Required the side of a square bar of wrought iron, capable of re- 
sisting the twist of 600 lbs. on the end of a lever 8 feet long. 
600 X 96 
^ — 51^0 — ~ ^i inches. 



STRENGTH OP MATERIALS. 



28T 



In the case of revolving shafts for machinery, &c., the strength 
is directly as the cubes of their diameters, and revolutions, and in- 
versely as the resistance they have to overcome ; hence, 

From practice, we find that a 40 horse power steam engine, 
making 25 revolutions per minute, requires a shaft (if made of 
wrought-iron) to be 8 inches diameter : now, the cube of 8, multi- 
plied by 25, and divided by 40 = 320 ; which serves as a constant 
multiplier for all others in the same proportion. 

What must be the diameter of a wrought iron shaft for an engine 
of 65 horse power, making 23 revolutions per minute ? 



^- 



65 X 320 
23 



9*67 inches diameter. 



James Glenie, the mathematician, gives 400 as a constant mul- 
tiplier for cast iron shafts that are intended for first movers in ma- 
chinery ; 

200 for second movers ; and 
100 for shafts connecting smaller machinery, &c. 
The velocity of a 30 horse power steam engine is intended to be 
19 revolutions per minute. Required the diameter of bearings for 
the fly-wheel shaft. 



^' 



400 X 30 



19 



= 8-579 inches diameter. 



Required the diameter of the bearings of shafts, as second movers 
from a 30 horse engine ; their velocity being 36 revolutions per 
minute. 



^- 



20 X 30 
36 



= b'b inches diameter. 



When shafting is intended to be of wrought iron, use 160 as the 
multiplier for second movers ; and 80 for shafts connecting smaller 
machinery. 

Table of the Proportionate Length of Bearings, or Journals for 

Shafts of various diameters. 



Dia. ia Inches. 


Len. ia Inches. 


Dia. in Inches. 


Len. in Inches. 


1 


If 


6i 


8| 


IJ 


2i 


7 


H 


2 


8 


7i 


10 


2i 


Si 


8 


lOf 


^ 


^ 


8* 


llf 


3 


H 


9 


12 


8J 


4| 


n 


12f 


4 


5J 


10 


13i 


^ 


H 


101 


14 


5 


6-i 


11 


lU 


5J 


n 


11} 


151 


6 


H 


12 


16 



288 



THE PRACTICAL MODEL CALCULATOR. 



Tenacities, Resistances to Compression, and other Properties of the 
common Materials of Construction. 



Names of Bodies. 


Absolute. 


Compared -with Cast Iron. 


Tenacity in lbs. 
per sq. inch. 


Resistance to 

compression 

in lbs. per sq. 

in. 


Its strength 
is 


Its extensi- 
bility is 


Its stiffness is 


Ash 


14130 
12225 
17968 
275 
13434 
33000 
9720 
12346 
11800 
11835 

35838 

56000 

12240 

1824 

11475 

551 

11880 

200 

128000 

478 

772 

2661 

857 

4736 

9120 


8548 

10304 

562 

86397 

1033 
2028 
5375 
5445 
10910 

5568 

8000 
6060 
9504 

5490 
6630 
3729 


0-23 
0-15 
0-435 

1-000 

0-21 
0-23 
0-3 
0-25 

0-65 

1-12 

0-136 

0-096 

0-24 

0-25 

0-182 
0-365 


2-6 
2-1 
0-9 

1-0 

2-9 
2-4 
2-4 
2-9 

1-25 

0-86 

2-3 

2-5 

2-9 

2-8 

0-75 
0-5 


0-089 
0-073 
0-49 

1-000 

0-073 
0-1 
0-1 
0-087 

0-535 

1-3 

0-058 

0-0385 

0-487 

0-093 

0-25 
0-76 


Beech 


Brass 


Brick 


"Cast iron 


Copper (wrought) 

Elm 


Fir, or Pine, white 

— — red 

— — yellow.... 

Granite, Aberdeen 

Gun-metal (copper 8, 

and tin 1) 




Larch 


Lead 


Mahogany, Honduras.. 
Marble 


Oak 

Rope (1 in. in circum.) 
Steel 


Stone, Bath 


— Craigleith 

— Dundee 

— Portland 


Tin (cast) 

Zinc (sheet) 



Comparative Strength and Weight of Hopes and Chains. 



1 


u 

n 


ii 


^1 


Proof 
strength in 
tons & cwt. 


2 

il 


II 

II 


j! 

II 


II 


Proof strength 
in tons & owt. 


^ 


2f 


^ 


5^ 


1 51 


10 


23 


7 
8 


43 


10 


4i 


4| 


1 


8 


1 16| 


lOf 


28 


M 


49 


11 11 


5 


5i 


t'b 


101 


2 10 


111 


30* 


lin. 


56 


13 8 


5f 


7 


^ 


14 


3 51 


m 


36 


It^ 


63 


14 18 


^ 


9f 


1^ 


18 


4 31 


13 


39 


1^ 


71 


16 14 


7 


11* 


t 


22 


5 2 


13f 


45 


li% 


79 


18 11 


8 


15 


H 


2T 


6 41 


141 


48^ 


1{ 


87 


20 8 


8| 


19 


f 


32 


7 7 


151 


56 


H 


96 


22 13 


n 


21 


M 


37 


8 13i 


16 


60 


If 


106 


24 18 



It must be understood and also borne in mind, that in estimating 
tbe amount of tensile strain to which a body is subjected, the weight 
of the body itself must also be taken into account ; for according 
to its position so may it approximate to its whole weight, in tend- 



STRENGTH OF MATERIALS. 



289 



ing to produce tension within itself; as in the almost constant 
application of ropes and chains to great depths, considerable 
heights, &c. 

Alloys that are of greater Tenacity than the sum of their Constitu- 
ents, as determined by the Experiments of Muschenhroek. 

Swedish copper 6 parts, Malacca tin 1 — tenacity per square inch 64,000 lbs. 

Chili copper 6 parts, Malacca tin 1 60,000 

Japan copper 5 parts, Banca tin 1 57,000 

Anglesea copper 6 parts, Cornish tin 1 41,000 

Common block tin 4, lead 1, zinc 1 ^ 13,000 

Malacca tin 4, regulus of antimony 1 , 12,000 

Block tin 3, lead 1 10,200 

Block tin 8, zinc 1 10,000 

Lead 1, zinc 1 4,500 

TabIe of Bata^ containing the Results of Experiments on the Elas- 
ticity and Strength of various Species of Timber. 



Species of Timber. 



Value of E. 


Value of S. 


174-7 


2462 


122-26 


2221 


105 


1672 


155-5 


1766 


86-2 


1457 


70-5 


1383 


119 


2026 


98 


1556 



Species of Timber. 



Value of E. Value of S. 



Teak 

Poona 

English oak. . 
Canadian do. 
Dantzic do. 
Adriatic do. 

Ash 

Beech 



Elm 

Pitch pine 

Red pine 

New England fir 

Riga fir 

Mar Forest do. 

Larch 

Norway spruce... 



50-64 

88-68 
133 
158-5 

90 

63 

76 
105-47 



1013 
1632 
1341 
1102 
1100 
1200 
900 
1474 



Rule. — To fi7id the dime7isions of a beam capable of sustaining 
a given weight, with a given degree of deflection, when supported 
at both ends. — Multiply the weight to be supported in lbs. by the 
cube of the length in feet ; divide the product by 32 times the 
tabular value of E, multiplied into the given deflection in inches, 
and the quotient is the breadth multiplied by the cube of the depth 
in inches. 

When the beam is intended to be square, then the fourth root 
of the quotient is the breadth and depth required. 

If the beam is to be cylindrical, multiply the quotient by 1*7, 
and the fourth root of the product is the diameter. 

The distance between the supports of a beam of Riga fir is 
16 feet, and the weight it must be capable of sustaining in the 
middle of its length is 8000 lbs., with a deflection of not more 
than f of an inch ; what must be the depth of the beam, suppos- 
ing the breadth 8 inches ? 

16 X 8000 

90 X 32 X -75 ^ ^^-^'^^ ^^^ "^^^^ ' "" ^^'^^ '^^' ^^'^ ^^i'^^'- 
Rule. — To determine the absolute strength of a recta^igular beam 
of timber when supported at both ends, and loaded in the middle 
of its length, as beams in general ought to be calculated to, so that 
they may be rendered capable of withstanding all accidental casea 
of emergency. — Multiply the tabular vs^lue of S by four times the 
depth of the beam in inches, and by the area of the cross section 
in inches ; divide the product by the distance between the supports 
Z 19 



290 THE PRACTICAL MODEL CALCULATOR. 

in inches, and the quotient Tvill be the absolute strength of the 
beam in lbs. 

If the beam be not laid horizontally, the distance between the 
supports, for calculation, must be the horizontal distance. 

One-fourth of the weight obtained bj the rule is the greatest 
weight that ought to be applied in practice as permanent load. 

If the load is to be applied at any other point than the middle, 
then the strength will be, as the product of the two distances is to 
the square of half the length of the beam between the supports; 
or, twice the distance from one end, multiplied by twice from the 
other, and divided by the whole length, equal the effective length 
of the beam. 

In a building 18 feet in width, an engine boiler of 5J tons is to 

be fixed, the centre of which to be T feet from the wall ; and having 

two pieces of red pine 10 inches by 6, which I can lay across the 

two walls for the purpose of slinging it at each end, — may I with 

sufficient confidence apply them, so as to effect this object ? 

2240 X 5-5 ^ , , 
-^ = 6160 lbs. to carry at each end. 

14 X 22 

And 18 feet — 7 = 11, double each, or 14 and 22, then — ^g — 

= 17 feet, or 204 inches, effective length of beam, 
m , , nc , . 1-341 X 4 X 10 X 60 ^ ^^^^ 
Tabular value of S, red pme = oqI ~ 15776 

lbs., the absolute strength of each piece of timber at that point. 

Rule. — To determine the di7nen§ions of a rectangular beam capa- 
ble of supporting a required weight, ivith a given degree of deflection, 
ivhen fixed at one end. — Divide the weight to be supported, in lbs., 
by the tabular value of E, multiplied by the breadth and deflection, 
both in inches ; and the cube root of the quotient, multiplied by 
the length in feet, equal the depth required in inches. 

A beam of ash is intended to bear a load of 700 lbs. at its ex- 
tremity ; its length being 5 feet, its breadth 4 inches, and the de- 
flection not to exceed ^ an inch. 

Tabular value of E = 119 X 4 X -5 = 238, the divisor ; then 
700 -^ 238 = ^2^ X 5 = 7*25 inches, depth of the beam. 

Rule. — To find the absolute strength of a rectangular beam, when 
fixed at one end, and loaded at the other. — Multiply the value of S 
by the depth of the beam, and by the area of its section, both in 
inches ; divide the product by the leverage in inches, and the quo- 
tient equal the absolute strength of the beam in lbs. 

A beam of Riga fir, 12 inches by 4|^, and projecting 6 J feet from 
the wall ; what is the greatest weight it will support at the ex- 
tremity of its length ? 

Tabular value of S = 1100 
12 X 4*5 == 54 sectional area, 

1100 X 12 X 54 ^ ^^ „ 
Then, fo = 9138-4 lbs. 



STRENGTH OF MATERIALS. 291 

When fracture of a beam is produced by vertical pressure, the 
fibres of the lower section of fracture are separated by extension, 
whilst at the same time those of the upper portion are destroyed 
by compression ; hence exists a point in section where neither the 
one nor the other takes place, and which is distinguished as the 
point of neutral axis. Therefore, by the law of fracture thus esta- 
blished, and proper data of tenacity and compression given, as in 
the Table (p. 281), we are enabled to form metal beams of strongest 
section with the least possible material : thus, in cast iron the re- 
sistance to compression is nearly as 6| to 1 of tenacity ; conse- 
quently a beam of cast iron, to be of strongest section, must be 
of the form TB, and a parabola in the direction of its 
length, the quantity of material in the bottom flange 
being about 6 J times that of the upper : but such is not 
the case with beams of timber; for although the tenacity 
of timber be on an average twice that of its resistance to compres- 
sion, its flexibility is so great, that any considerable length o£beam, 
where columns cannot be situated to its support, requires to be 
strengthened or trussed by iron rods, as in the following manner : 



i 




And these applications of principle not only tend to diminish de- 
flection, but the required purpose is also more efl'ectively attained, 
and that by lighter pieces of timber. 

Rule. — To ascertain the absolute strength of a east iron beam of 
the preceding form^ or that of strongest section. — Multiply the sec- 
tional area of the bottom flange in inches by the depth of the beam 
in inches, and divide the product by the distance between the sup- 
ports also in inches ; and 514 times the quotient equal the absolute 
strength of the beam in cwts. 

The strongest form in which any given quantity of matter can 
be disposed is that of a hollow cylinder ; and it has been demon- 
strated that the maximum of strength is obtained in cast iron, when 
the thickness of the annulus or ring amounts to |th of the cylinder's 
external diameter ; the relative strength of a solid to that of a 
hollow cylinder being as the diameters of their sections. 

The following table shows the greatest weight that ever ought 
to be laid upon a beam for permanent load, and if there be any 
liability to jerks, &c., ample allowance must be made ; also, the 
weight of the beam itself must be included. 

Rule. — To find the weight of a cast iron beam of given dimen- 
sions. — Multiply the sectional area in inches by the length in feet, 
and by 3*2, the product equal the weight in lbs. 

Required the weight of a uniform rectangular beam of cast iron, 
16 feet in length, 11 inches in breadth, and 1|- inch in thickness. 
11 X 1-5 X 16 X 3-2 = 844-8 lbs. 



292 



THE PRACTICAL MODEL CALCULATOR. 



A Table showing the Weight or Pressure a Beam of Cast Iron, 
1 i7ich in breadth, will sustain without destroying its elastic force, 
when it is supported at each end, and loaded in the middle of its 
length, and also the deflection in the middle which that weight 
will produce. 



Length. 


6 feet. 


7 feet. 


8 feet. 


9 feet. 


10 feet. 1 


Depth 


Wt. in 


Defl. in 


Wt. in 


Defl. in 


Wt. in 


Defl. in 


Wt.in 


Defl. in 


Wt. in 


Defl. in 


in in. 


lbs. 


in. 


lbs. 


in. 


lbs. 


in. 


lbs. 


in. 

•54 


lbs. 

765 


in. 

•66 


3 


1278 


•24 


1089 


•33 


954 


•426 


855 


3i 


1789 


•205 


1482 


•28 


1298 


•365 


1164 


•46 


1041 


•57 


4 


2272 


•18 


1936 


•245 


1700 


•32 


1520 


•405 


1360 


•5 


^ 


2875 


•16 


2450 


•217 


2146 


•284 


1924 


•36 


1721 


•443 


5 


3560 


•144 


3050 


•196 


2650 


•256 


2375 


•32 


2125 


•4 


6 


5112 


•12 


4356 


•163 


3816 


•213 


3420 


•27 


3060 


•33 


7 


6958 


•103 


5929 


•14 


5194 


•183 


4655 


•23 


4165 


•29 


8 


9088 


•09 


7744 


•123 


6784 


•16 


6080 


•203 


5440 


•25 


9 








9801 


•109 


8586 


•142 


7695 


•18 


6885 


•22 


10 


• — 





12100 


•098 


10600 


•128 


9500 


•162 


8500 


•2 


11 













12826 


•117 


11495 


•15 


10285 


•18 


12 














15264 


•107 


13680 


•135 


12240 


•17 


13 * 




















16100 


•125 


14400 


•154 


14 

6 


— 


— 


— 


~ 




— 


18600 


•115 


16700 


•143 


12 feet. 


14 feet. 


16 feet. 


18 fe 


et. 


20 feet. 1 


2548 


•48 


2184 


•65 


1912 


•85 


1699 


1^08 


1530 


1-34 


7 


3471 


•41 


2975 


•58 


2603 


•73 


2314 


•93 


2082 


1-14 


8 


4532 


•36 


3884 


•49 


3396 


•64 


3020 


•81 


2720 


1-00 


9 


5733 


•32 


4914 


•44 


4302 


•57 


3825 


•72 


3438 


•89 


10 


7083 


•28 


6071 


•39 


5312 


•51 


4722 


•64 


4250 


•8 


11 


8570 


•26 


7346 


•36 


6428 


•47 


6714 


•59 


5142 


•73 


12 


10192 


•24 


8736 


•33 


7648 


•43 


6796 


•54 


6120 


•67 


13 


11971 


•22 


10260 


•31 


8978 


•39 


7980 


•49 


7182 


•61 


14 


13883 


•21 


11900 


•28 


10412 


•36 


9255 


•46 


8330 


•57 


15 


15937 


•19 


13660 


•26 


11952 


•34 


10624 


•43 


9562 


•53 


16 


18128 


•18 


35536 


•24 


13584 


•32 


12080 


•40 


10880 


•5 


17 


20500 


•17 


17500 


•23 


15353 


•3 


13647 


•38 


12282 


•47 


18 


22932 


•16 


19656 


•21 


17208 


•28 


15700 


•36 


13752 


•44 



Resistance of Bodies to Flexure hy Vertical Pressure. — When a 
piece of timber is employed as a column or support, its tendency 
to yielding by compression is different according to the proportion 
between its length and area of its cross section ; and supposing the 
form that of a cylinder whose length is less than seven or eight times 
its diameter, it is impossible to bend it by any force applied longi- 
tudinally, as it will be destroyed by splitting before that bending 
can take place ; but when the length exceeds this, the column will 
bend under a certain load, and be ultimately destroyed by a similar 
kind of action to that which has place in the transverse strain. 

Columns of cast iron and of other bodies are also similarly cir- 
cumstanced. 

When the length of a cast iron column with flat ends equals 
about thirty times its diameter, fracture will be produced wholly by 
bending of the material ; — when of less length, fracture takes place 
partly by crushing and partly by bending : but, when the column 



STRENGTH OF MATERIALS. 



293 



is enlarged in the middle of its length from one and a half to twice 
its diameter at the ends, by being cast hollow, the strength is 
greater by ith than in a solid column containing the same quantity 
of material. 

Rule.' — To determine the dimensions of a support or column to 
hear without sensible curvature a given p>ressure in the directioii of 
its axis. — Multiply the pressure to be supported in lbs; by the 
square of the column's length in feet, and divide the product by 
twenty times the tabular value of E ; and the quotient will be equal 
to the breadth multiplied by the cube of the least thickness, both 
being expressed in inches. 

When the pillar or support is a square, its side will be the fourth 
root of the quotient. 

If the pillar or column be a cylinder, multiply the tabular 
value of E by 12, and the fourth root of the quotient equal the 
diameter. 

What should be the least dimensions of an oak support, to bear 
a weight of 2240 lbs. without sensible flexure, its breadth being 3 
inches, and its length 5 feet ? 

2240 X 52 



Tabular value of E = 105, and 2 x 105 x 3 "^ ^8-888 = 
2*05 inches. 

Required the side of a square piece of Riga fir, 9 feet in length, 
to bear a permanent weight of 6000 lbs. 

6000 X 9- 

Tabular value of E = 96, and qq ^ qn — ''\/253 = 4 inches 

nearly. 

Dimensions of Cylindrical Columns of Cast Iron to sustain a given 
load or 'pressure with safety. 



11 

5 










Length or heigl 


t in feet. 










4 


6 1 


8 1 


10 


12 


14 


16 


1- 18 


1 20 


1 22 


1 24 










Weight or load 


iu cvHs. 










2 


72 


60 


49 


40 


32 


26 


22 


18 


15 


13 


11 


u 


119 


105 


91 


77 


65 


•55 


47 


40 


34 


29 


25 


3 


178 


163 


145 


128 


111 


97 


84 


73 


64 


56 


49 


3* 


247 


232 


214 


191 


172 


156 


135 


119 


106 


94 


83 


4 


326 


310 


288 


266 


242 


220 


198 


178 


160 


144 


130 


^ 


418 


400 


379 


354 


327 


301 


275 


251 


229 


208 


189 


r> 


522 


501 


479 


452 


427 


394 


365 


337 


310 


285 


262 


6 


607 


692 


573 


550 


525 


497 


469 


440 


413 


386 


360 


7 


1032 


1013 


989 


959 


924 


887 


848 


808 


765 


725 


686 


8 


1333 


1315 


1289 


1259 


1224 


1185 


1142 


1097 


1052 


1005 


959 


9 


1716 


1697 


1672 


1640 


1603 


1561 


1515 


1467 


1416 


1364 


1311 


10 


2119 


2100 


2077 


2045 


2007 


1964 


1916 


1865 


1811 


1755 


1697 


11 


2570 


2550 


2520 


2490 


2450 


2410 


2358 


2805 


2248 


2189 


2127 


12 


3050 


3040 


3020 


2970 


2930 


2900 


2830 


2780 


2730 


2670 


2600 



Practical utility of the preceding Table. — Wanting to support the 
front of a building with cast iron columns 18 feet in length, 8 inches 
in diameter, and the metal 1 inch in thickness ; what weight may 
z2 



294 THE PRACTICAL MODEL CALCULATOR. 

I confidently expect each column capable of supporting without 

tendency to deflection ? 

Opposite 8 inches diameter and under 18 feet = 1097 
Also opposite 6 in. diameter and under 18 feet = 440 

= 657 cwts. 
The strength of cast iron as a column being = 1-0000 

— steel — = 2-518 

— wrought iron — = 1-745 

— oak (Dantzic) — = -1088 

— red deal — = -0785 
Elasticity of torsion, or resistance of bodies to twisting. — The 

angle of flexure by torsion is as the length and extensibility of the 
body directly, and inversely as the diameter ; hencc. the length 
of a bar or shaft being given, the power, and the leverage the 
power acts with, beir.c ^P'^wn, and also the number of degrees of 
torsion that will not ^ .cct the action of the machine, to determine 
the diameter in cast iron with a given angle of flexure. 

Rule. — Multiply the power in lbs. by oI.d length of the shaft in 
feet, and by the leverage in feet ; divide the product by fifty-five 
times the number of degrees in the angle of torsion, and the fourth 
root of the quotient equal the shaft's diameter in inches. 

Required the diameters for a series of shafts 35 feet in length, 
and to transmit a power equal to 1245 lbs., acting at the circum- 
ference of a wheel 2J feet radius, so that the twist of the shafts 
on the application of the power may not exceed one degree. 

1245 X 35 X 2-5 -_ _ , ,^ . , . ,. 
rr ^ 1 = '^v/lyol = 0-67 inches m diameter. 

Relative strength of metals to resist torsion. 

Cast iron = 1 Swedish bar iron ...= 1*05 

Copper = -48 English do = 1-12 

Yellow brass = -511 Shear steel = 1-96 

Gun-metal = -55 Cast do = 2*1 

Deflexion of Rectangular Beams. 

Rule. — To ascertain the amount of deflexion of a uniform heam 
of east iron, supported at hoth ends, and loaded in the middle to the 
extent of its elastic force. — Multiply the square of the length in feet 
by -02, and the product divided by the depth in inches equal the 
deflexion. 

Required the deflection of a cast iron beam 18 feet long between 
the supports, 12-8 inches deep, 2*56 inches in breadth, and bear- 
ing a weight of 20,000 lbs. in the middle of its length. 

— YoTo — = *506 inches from a straight line in the middle. 

For beams of a similar description, loaded uniformly, the rule is 
the same, only multiply by -025 in place of -02. 

Rule. — To find the deflection of a beam when fixed at one end 



STRENGTH OF MATERIALS. 



295 



and haded at the other. — Divide the length in feet of the fixed part 
of the beam by the length in feet of the part which yields to the 
force, and add 1 to the quotient ; then multiply the square of the 
length in f^et by the quotient so increased, and also by '13 ; divide 
this product by the middle depth in inches, and the quotient will 
be the deflection, in inches also. 

Multiply the deflection so obtained for cast iron by -86, the pro- 
duct equal the deflection for wrought iron ; for oak, multiply by 
2-8 ; and for fir, 2-4, 

A Table of the Depths of Square Beams or Bars of Cast Iron^ 
calculated to support from 1 Cwt, to 14 Toris in the Middle^ the 
Deflection not to exceed ^^th of an Inch for each Foot in Length. 



LeoCthB 


9 fleet 


4 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 


26 


28 

t 


30 

0, 


Weight ia 


We^tin 


i 


i 


"t" 


~i' 


t. 


~~2'~ 

■5. 


1 


In. 


i 


■£ 


<wt- 


1" 


_«_ 


(S 


a 


_«_ 


a 


a 


In. 


In. 


In. 


In. 


In. 


In. 






In. 


In. 


In. 


In. 


In. 


In. 


In. 


Icwt. 


112 


1-2 


1-4 


1-7 


1-9 


2-0 


2-2 


2-4 


2-5 


2^6 


2-7 


2-9 


30 


3-1 


3-2 


2 


124 


1-4 


1-7 


2-0 


2-2 


2-4 


2-6 


2-8 


3^0 


31 


3-3 


3-4 


3-6 


37 


3-S 


3 


336 


1-6 


1-9 


2-2 


2-4 


2-7 


2-9 


3-1 


33 


3^4 


3-6 


3-8 


3-9 


41 


4-2 


4 


448 


1-7 


2-0 


2-4 


2-6 


2-9 


31 


3-3 


3-5 


3-7 


3^9 


4-0 


4-2 


4-3 


4-5 


5 


560 


1-8 


2-2 


2-5 


2-8 


30 


3-3 


3-5 


3-7 


3-9 


4-1 


4-3 


4-4 


4-6 


4-8 


6 


672 


1-8 


2-2 


2-6 


2-9 


3-2 


3-4 


3-7 


3-9 


4-1 


4-3 


4-5 


4-6 


4-8 


50 


7 


784 


1-9 


2-3 


2-7 


3-0 


3-3 


3-6 


3-8 


41 


4-2 


4.4 


4-6 


4-8 


50 


5^2 


8 


896 


2-0 


2-4 


2-8 


3-1 


3-4 


3-7 


3-y 


4-2 


4-4 


4-6 


4-8 


5-0 


5-2 


54 


9 


1,008 


2-0 


2-5 


2-9 


3-2 


3-5 


3-8 


4-0 


4-3 


4-5 


4^7 


4-9 


5-1 


6-3 


5^5 


10 


1,130 


2-1 


2-6 


3-0 


3-3 


3-6 


3-9 


4-2 


4-4 


4-7 


4^9 


5-2 


6-3 


6-4 


5-7 


11 


1,232 


2-1 


26 


3-0 


34 


3-7 


4-0 


4-3 


4-5 


4-8 


5-0 


5-3 


5-4 


5-6 


5-8 


12 


1,344 


2-2 


27 


3-1 


3-5 


3-8 


4-1 


4-4 


4-7 


4-9 


5-1 


5-3 


5-5 


5-7 


6-9 


13 


1,456 


2-2 


2-7 


3-1 


3-5 


3-8 


4-2 


4-4 


4-7 


4-9 


5-2 


5-4 


5-6 


5-9 


60 


14 


1,568 


2-3 


2-8 


3-2 


3-6 


3-9 


4-2 


4-5 


4-8 


50 


5-3 


5-5 


5-7 


6-0 


6-1 


15 


1,680 


2-3 


2-8 


3-2 


3-6 


40 


4-3 


4-6 


4-9 


5-2 


5-4 


5-6 


5-8 


6-1 


6-2 


16 


1,792 


2-4 


2-9 


3-3 


3-7 


40 


4-4 


4-7 


5-0 


5-2 


5-5 


5-7 


5-9 


6-2 


6-4 


17 


1,904 


2-4 


2-9 


3-4 


3-8 


41 


4-4 


4-7 


5-0 


5-3 


5-5 


5-8 


fro 


6-2 


6-5 


18 


2,016 


2-4 


30 


3-4 


3-8 


4-2 


4-5 


4-8 


5-1 


5-4 


56 


5-9 


61 


6-4 


6-6 


19 


2,128 


2-5 


30 


3-5 


3-9 


4-2 


4-6 


4-9 


5^2 


5-4 


5-7 


6-0 


6-2 


6-0 


6-7 


1 too. 


2,240 


2-5 


3-0 


3-5 


3-9 


4-3 


4-6 


4-9 


5-2 


5-5 


5-8 


6-0 


6-3 


6-5 


6-8 


U 


2,800 


2-6 


3-2 


3-7 


41 


4-5 


4-9 


5-2 


5-5 


5^8 


6-1 


6-4 


6-6 


6-9 


7-2 


li 


3,360 


2-8 


3-4 


3-9 


4-3 


4-7 


51 


5-5 


5-8 


6-1 


6-4 


6-7 


70 


7'2 


7^5 


u 


3,920 


2-9 


3-5 


4-0 


4-5 


4-9 


5-3 


5-7 


6-0 


6-3 


67 


6-9 


7-2 


I'a 


7-7 


2 


4,480 


2-9 


3-5 


41 


4-7 


5-1 


5-5 


5-9 


6-2 


6-5 


6-8 


7-2 


7-6 


"i'l 


8-0 


2i 


6,600 


3-1 


3-8 


4-4 


4-9 


5-5 


5-8 


6-2 


6-6 


6^9 


7-3 


7-6 


7-9 


8^2 


8-0 


3 


6,720 


3-3 


4-0 


4-6 


61 


5-7. 


61 


6-5 


6-9 


7-3 


7-6 


7-9 


8-3 


8-6 


8-9 


3^ 


7,540 


3-4 


4-1 


4-8 


5-3 


5-8 


6-3 


6-7 


7-1 


7-5 


7-9 


8-2 


8-6 


8-9 


9-2 


4 


8,960 


3-5 


4-3 


4-9 


5-5 


6-0 


6-5 


7-0 


7-4 


7-8 


8-2 


8-5 


8-9 


9-2 


9-5 


4i 


10,080 


— 


4-4 


51 


5-7 


6-2 


6-7 


7-2 


7-6 


80 


8-4 


8-8 


9-1 


9-5 


9-8 


5 


11,200 





4-5 


5-2 


5-8 


6-4 


6-9 


7-4 


7-8 


8-2 


S-6 


90 


9-4 


9-7 


10-1 


6 


13,440 








6-5 


6-1 


6-7 


7-2 


7-7 


8-2 


8-6 


9-0 


9-4 


9-8 


10-2 


10-5 


7 


15,680 


— 


— 


5-7 


6-3 


6-9 


7-5 


8-0 


8-5 


8-9 


9-4 


9-8 


10^2 


10-6 


11^0 


8 


17,920 


— 


— 


5-9 


6-6 


7-2 


7-8 


8-3 


8-8 


9-3 


9-7 


10-1 


10^6 


10-9 


11-3 


9 


20,160 


— 




6-0 


6-8 


7-4 


8-0 


8-5 


9-0 


9-5 


100 


10-4 


10-9 


11^3 


11-7 


10 


22,400 











6-9 


7-6 


8-2 


8-8 


9-3 


9^8 


10-3 


10-7 


11-2 


11-6 


120 


11 


24,640 


— 








7-1 


7-8 


8-4 


90 


95 


10-0 


10-5 


11-0 


11^5 


11-9 


12-3 


12 


26,880 


— 


— 


— 


7-2 


7-9 


8-6 


9-2 


9-7 


102 


10-8 


11-2 


117 


12-1 


12-5 


13 


29,120 


— 




— 


7-4 


8-1 


8-8 


9-4 


9-9 


10-4 


11-0 


11-5 


11-9 


12-4 


12-8 


14 


31,360 


— 


— 


— 


7-5 


8-3 


8-9 


9-5 


10-1 


10-6 


11-1 


11-7 


12^1 


12-6 
-7 


13-0 
~b 


Deflection 


in iachei 


•1 


•15 


•2 


•25 


•3 


•35 


'~^ 


-45 


'b 


~55" 


""^6" 


•65 


Lengths 


in Feet 


10 


12 


14 


16 


18 


20 


22 


J!- 


26 


28 


30 


32 


34 


~^' 


15 


33,600 


7-7 


8-4 


91 


9-7 


10-3 


10-8 


11-4 


11-9 


12-3 


12-8 


13-2 


13-7 


14-1 


14-5 


16 


35,840 


7-8 


8-5 


9-2 


9-8 


10-4 


11^0 


11-5 


12-0 


12-5 


13-0 


13-5 


13^9 


14-3 


147 


17 


38,080 


7-9 


87 


9-4 


10-0 


10-6 


112 


11-7 


12^2 


12^7 


13-2 


13-7 


141 


14-5 


14^9 


18 


40.320 


80 


8-8 


9-5 


101 


10-8 


11-3 


11-9 


12-4 


12-9 


13-4 


13-9 


14^3 


14-7 


151 


19 


42,5'iO 


8-1 


8-9 


9-6 


10-3 


10-9' 


11-5 


12-2 


12-6 


13-1 


13-6 


14-1 


14-5 


150 


154 


20 


44,800 


— 


9-0 


9-7 


10-4 


11-0 


116 


125 


1-2-7 


13-2 


13-8 


14-2 


14-7 


151 


15^6 


22 


49,280 





9-2 


100 


10-7 


11-3 


119 


12-8 


13-0 


13-6 


14-1 


14-6 


151 


15-5 


15^9 


24 


53,760 


— 


9-4 


10-2 


10-9 


11-5 


12-2 


130 13-4 


13-9 


14-4 


14-9 


15-4 


15-9 


16-3 


26 


58,240 





9-6 


10-4 


11-1 


11-8 


12-4 


13-3 13-6 


142 


14-7 


15-2 


15-7 


16-2 


16-7 


28 


62,720 


— 


9-8 


10-6 


11-4 


12-0 


12-7 


13-5 13-9 


14-4 
•66 


150 


15-5 
•75 


160 
•8 


l&b 
•85 


17-0 
~9 


Dcflestion 


■D inches 


~25 


~3' 


~-35 


•4 1 -45 j 


•6 


•55 -6 



296 



THE PRACTICAL MODEL CALCULATOR. 



Length 


in Feet 


14 


16 


18 


20 


22 


24 


26 


28 


30 


32 


34 


36 


38 


40 


Weight in 


'Weight in 


U 


i 


i 


t 


i 




i 




t 


p. 


i. 


i. 


t 


f, 


tons. 




« 


fl 


ft 


ft 


R 


ft 


s 


n 


& 


p 


ft 


& 


A 


ft 






In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


30 


67,200 


10-8 


11-5 


12-2 


12^9 


13-5 


14-1 


14^7 


15^2 


15-7 


16-3 


16-8 


17-3 


17-7 


18-2 


32 


71,680 


11-0 


11-7 


12-4 


13-1 


l:!-7 


14-3 


14-9 


15-5 


16-0 


16-5 


170 


17-5 


18-0 


18-5 


34 


76,160 


11-1 


11-9 


12-6 


13-3 


13-9 


14-5 


15-1 


15-7 


16-2 


16-8 


17-3 


17-8 


18-3 


18-S 


36 


80,640 


11-3 


12-0 


12-8 


13-4 


14-1 


14-7 


15-3 


15-9 


16-5 


17-0 


17-5 


18^0 


18-5 


19-0 


38 


85,120 


11-4 


12-2 


13-0 


13-6 


14-3 


14-9 


15-5 


16-1 


16-7 


17-2 


17-S 


18-3 


18-8 


19-3 


40 


89,600 





12-4 


13-1 


13^8 


14-5 


15^1 


15-7 


16-4 


16-9 


17-5 


18-0 


18-5 


19-1 


19-5 


42 


94,080 





12-5 


13-3 


14-0 


14-7 


15-3 


15-9 


16-5 


17-1 


17-7 


lS-2 


18^7 


19-3 


19-8 


44 


98,560 





12-7 


13-5 


14-2 


14-9 


15-5 


16-1 


16-8 


17-4 


17-9 


18-5 


19-0 


19-5 


20-0 


46 


103,040 





12-8 


13-6 


U-3 


15-0 


15-7 


16^3 


17-0 


17-6 


18-1 


18-7 


19-2 


19'8 


20-3 


48 


107.520 


— 


13-0 


1.3-7 


14-5 


15-2 


15-9 


16-5 


17-1 


17-7 


18-3 


18-8 


19^4 


20-0 


20-5 


50 


112.000 








13-8 


14-6 


lo-3 


IfrO 


16-6 


17-3 


17-9 


18-5 


19-0 


19-6 


■20-1 


207 


52 


116,480 








14-0 


14-7 


15-5 


16-2 


16-8 


17-5 


18-1 


18^7 


19-2 


19-8 


20-3 


21-0 


54 


120,960 








14-1 


U-9 


15-7 


16-3 


17^0 


17-6 


18-2 


18-8 


19-4 


19-9 


20-5 


2M 


56 


125,440 


— 





14-3 


15-0 


15-8 


16-5 


^•1 


17-8 


18-4 


190 


19-6 


201 


20-7 


21-3 


58 


129,920 


— 


— 


144 


15-1 


15-9 


16-6 


17-3 


17-9 


18-5 


19^2 


19-7 


20-3 


20^9 


21-4 


60 


134,400 


— 


— 


14-6 


15-3 


160 


16-7 


17-4 
~6b 


18^1 


18^7 


19-3 


199 
^5" 


2(J-5 
~~9" 


21 •I 

•95 


21-6 
1-0 


Deflectio 


a in inches 


•35 


•4 


•45 


•55 


-6 


■7 


•75 


•8 



Examples illustrative of the Table. — 1. To find the depth of a 
rectangular bar of cast iron to support a weight of 10 tons in the 
middle of its length, the deflection not to exceed ^-^ of an inch per 
foot in length, and its length 20 feet, also let the depth be 6 times 
the breadth. 

Opposite 6 times the weight and under 20 feet in length is 15*3 
inches, the depth, and \ of 15'3 = 2*6 inches, the breadth. 

2. To find the diameter for a cast iron shaft or solid cylinder 
that will bear a given pressure, the flexure in the middle not to ex- 
ceed ^th of an inch for each foot of its length, the distance of the 
bearings being 20 feet, and the pressure on the middle equals 10 
tons. 

Constant multiplier 1-7 for round shafts, then 10 X 1*7 = 17. 
And opposite 17 tons and under 20 feet is 11*2 inches for the di- 
ameter. 

But half that flexure is quite enough for revolving shafts : hence 
17 X 2 = 34 tons, and opposite 34 tons is 13-3 inches for the di- 
ameter. 

3. A body 256 lbs. weight, presses against its horizontal sup- 
port, so that it requires the force of 52 lbs. to overcome its friction ; 
if the body be increased to 8750 lbs., what force will cause it to 
pass from a state of rest to one of motion ? 

52 
256 

.-. 8750 X 203125 = 1777*34375 lbs., the force required. 
This calculation is based upon the law, that friction is propor- 
tional to the normal pressure between the rubbing surfaces. Twice 
the pressure gives twice the friction ; three times the pressure gives 
three times the friction ; and so on. With light pressures, this law 
may not hold, but then it is to be attributed to the proportionately 
greater effect of adhesion. 

4. If a sleigh, weighing 250 lbs., requires a force of 28 lbs. to 
draw it along ; when 1120 lbs. are placed in it, required the units 
of work expended to move the whole 350 feet ? 



= '203125 = , in this case, the coefficient of friction; 



STRENGTH OF MATERIALS. 



297 



_28 
250 



= •112, the coefficient of friction. 



Then (1120 + 250) x -112 = 153-44 lbs., the force required to 
move the whole. 

.-. 153-44 X 350 = 53704, the units of work required. 

A UNIT OF WORK is the labour which is equal to that of raising 
one pound a foot high. It is supposed that a horse can perform 
33000 units of work in a minute. 

It may also be remarked that friction is independent of the ex- 
tent of the surfaces in contact, except with trifling pressures and 
large surfaces, which is on account of the effect of adhesion. The 
friction of motion is independent of velocity, and is generally less 
than that of quiescence. a 

5. Required the co- 
efficient of friction, for 
a sliding motion^ of 
cast iron upon wrought, 
lubricated with Dev- 
lin's oil, and under 
the following circum- 
stances : the load A, 
and sledge nm, weighs 
8420 lbs., and requires 
a weight W, of 1200 lbs. to cause it to pass from a state of rest 
into one of motion : the sledge and load pass over 22 feet on the 
horizontal way rs, in 8 seconds. 

In this case the coefficient of sliding motion will be 

1200 1200 + 8420 2 x 22 

8420 8420 ^ ^ x 8^' 

in which g = 32-2 feet ; the acceleration of the free descent of 

bodies brought about by gravity. The above expression becomes 

44 




•142515 - 1-142515 x 



2060- 



= -118121. 



Hence the coefficient of the friction of motion is -118121, and the 
coefficient of the friction of quiescence is -142515. 



BING ON EACH OTHER. 



In the years 1831, 1832, and 1833, a very extensive set of ex- 
periments were made at Metz, by M. Morin, under the sanction 
of the French government, to determine as nearly as possible the 
laws of friction ; and by which the following were fully established : 

1. When no unguent is interposed, the friction of any two sur- 
faces (whether of quiescence or of motion) is directly proportional 
to the force with which they are pressed perpendicularly together ; 
so that for any two given surfaces of contact there is a constant 
ratio of the friction to the perpendicular pressure of the one surface 
upon the other. Whilst this ratio is thus the same for the same 



298 THE PRACTICAL MODEL CALCULATOR. 

surfaces of contact, it is different for different surfaces of contact. 
The particular value of it in respect to any two given surfaces of 
contact is called the coefficient of friction in respect to those sur- 
faces. 

2. When no unguent is interposed, the amount o^ the friction is, 
in every case, wholly independent of the extent of the surfaces of 
contact ; so that, the force with which two surfaces are pressed to- 
gether being the same, their friction is the same, whatever may be 
the extent of their surfaces of contact. 

3. That the friction of motion is wholly independent of the velo- 
city of the motion. 

4. That where unguents are interposed, the coefficient of friction 
depends upon the nature of the unguent, and upon the greater or 
less abundance of the supply. In respect to the supply of the un- 
guent, there are two extreme cases, that in which the surfaces of 
contact are but slightly rubbed wdth the unctuous matter, as, for 
instance, with an oiled or greasy cloth, and that in which a con- 
tinuous stratum of unguent remains continually interposed between 
the moving surfaces ; and in this state the amount of friction is 
found to be dependent rather upon the nature of the unguent than 
upon that of the surfaces of contact. M. Morin found that with 
unguents (hog's lard and olive oil) interposed in a continuous stra- 
tum between surfaces of wood on metal, wood on wood, metal on 
wood, and metal on metal, when in motion, have all of them very 
near the same coefficient of friction, being in all cases included be- 
tween -07 and -08. 

The coefficient for the unguent tallow is the same, except in that 
of metals upon metals. This unguent appears to be less suited for 
metallic substances than the others, and gives for the mean value 
of its coefficient, under the same circumstances, 'lO. Hence, it is 
evident, that where the extent of the surface sustaining a given 
pressure is so great as to make the pressure less than that which 
corresponds to a state of perfect separation, this greater extent of 
surface tends to increase the friction by reason of that adhesiveness 
of the unguent, dependent upon its greater or less viscosity, whose 
effect is proportional to the extent of the surfaces between which 
it is interposed. 

It was found, from a mean of experiments with different unguents 
on axles, in motion and under different pressures, that, with the 
unguent tallow, under a pressure of from 1 to 5 cwt., the friction 
did not exceed ^th of the whole pressure ; when soft soap was ap- 
plied, it became g^th ; and with the softer unguents applied, such 
as oil, hog's lard, &c., the ratio of the friction to the pressure in- 
creased ; but with the harder unguents, as soft soap, tallow, and 
anti-attrition composition, the friction considerably diminished ; 
consequently, to render an unguent of proper efficiency, the nature 
of the unguent must be measured by the pressure or weight tend- 
ing to force the surfaces together. 



STRENGTH OF MATERIALS. 



299 



Table of the Results of Experiments on the Friction of Unctuous 
Surfaces. By M. Morin. 



Surfaces of Contact. 



Coef&cients of Friction. 



Friction of Friction of 
Motion. Quiescence. 



)ll, do., 
•ion, do. 



Oak upon oak, the fibres being parallel to the motion 
Ditto, the fibres of the moving body being perpendicu- 
lar to the motion 

Oak upon elm, fibres parallel 

Elm upon oak, do 

Beech upon oak, do 

Elm upon elm, do 

Wrought iron upon 
Ditto upon wrought 

lo 

ht iron, do 

•ass, do 

iron, dc ., 

Cast iron upon ojaL, do...... -. 

Ditto upon elm, do.^ tli3 uiiguont being tallow 

Ditto, do., the ungL-..:it being hog's lard and black 

lead 

Elm upon cast iron 

Cast iron upon cast iron , 

Ditto upon brass 

Brass upon cast iron 

Ditto upon brass 

Copper upon oak 

Yellow copper upon cast iron 

Leather (ox-hide), well tanned, upon cast iron, wetted 
Ditto upon brass, wetted 



Ditto upon cast iror 
Cast iron upon wrc 
Wrought iron upoi 
Brass upon wrou< 



0-018 

0-143 
0-136 
0-119 
0-330 
0-140 
0-138 
0-177 

0-143 
0-lGO 
0-166 
0-1G7 
0-125 

0-137 
0-135 
0-144 
0-132 
0-107 
0-134 
0-100 
0-115 
0-229 
0-244 



0-390 
0-314 
0-420 



0-118 
0-100 

0-098 

0-164 
0-267 



In these experiments, the surfaces, after having been smeared 
with an unguent, were wiped, so that no interposing layer of the 
unguent prevented intimate contact. 

Table of the Results of Experiments on Friction^ with Unguents 
interposed. By M. Morin. 



Surfaces of Contact 


Coefficients of Friction. 


Unguents. 


Friction of 
Motion. 


Friction of 
Quiescence. 


Oak upon oak, fibres parallel.... 
Do do 


0-164 
0-075 
0067 
0-083 
0-072 
0-250 
0-136 
0-073 
0-066 
0-080 
0-098 
0-055 
0-137 
0-170 
0-060 
0-139 
0-066 

0-256 

0-214 


0-440 
0-164 

0-254 
0-178 

o-iii 

0-142 
0-217 

0-649 


Dry soap. 
Tallow. 
Hog's lard. 
Tallow. 
Hog's lard. 
Water. 
Dry soap. 
Tallow. 
Hog's lard. 
Tallow. 
Tallow. 
Tallow. 
Dry soap. 
Tallow. 
Hog's lard. 
Dry soap. 
Tallow. 

f Greased and satu- 
t rated with water. 
Dry soap. 


Do do 




Do. do 


Do. do 


Do. upon elm, fibres parallel 

Do. do 


Do do 


Do upon cast iron • 


Do upon wrouffht iron.. 


Beech upon oak, fibres parallel.. 
Elm upon oak, do 


Do. do 


Do. do 


T'/liTi iirinn plm do 


Do upon cast iron 


Wrought iron upon oak, fibres \ 

parallel / 

Do do 





300 



THE PRACTICAL MODEL CALCULATOR. 



Surfaces of Contact. 



Wrouglit iron upon oak, fibres )^ 

parallel J 

Do. upon elm, do 

Do. do 

Do. do 

Do. upon cast iron, do 

Do. do 

Do. do 

Do. upon wrought iron, do 

Do. do 

Do. do 

Wrouglit iron upon brass, do 

Do. do 

Do. do 

Cast iron upon oak, do 



Do. 

Do. 

Do. 

Do. 

Do. upon elm, 

Do. 

Do. 



do. 

do 

do 

do, 

do, 

do 

do 



Do. upon wrought iron. 

Do. upon cast iron 

Do. do 

Do. do 

Do. do 

Do. do 



do. 



Do. 

Do. upon brass 

Do. do.... 

Do. do.... 

Copper upon oak, fibres parallel 

Yellow copper upon cast iron... 

Do. do 

Do. do.. 

Brass upon cast iron , 

Do. do 

Do. upon wrought iron.... 



do. 



Do. 

Do. do 

Brass upon brass 

Steel upon cast iron.. .. 

Do. do 

Do. do 

Do. upon wrought iron. 

Do.* do 

Do. upon brass 

Do. do 



Do. 



do. 



CocflScients of Friction. 



Friction of Friction of 
Motion. Quiescence. 



0-085 

0-078 
0-076 
0-055 
0-103 
0-076 
0-066 
0-082 
0-081 
0-070 
0-103 
0-075 
0-078 
0-189 

0-218 

0-078 
0-075 
0-075 
0-077 
0-061 

0-091 



0-314 
0-197 
0-100 
0-070 
0-064 

0-055 

0-103 
0-075 
0-078 
0-069 
0-072 
0-068 
0-066 
0-086 
0-077 
0-081 

0-089 

0-072 
0-058 
0-105 
0-081 
0-079 
0-093 
0-076 
0-056 
0-053 

0-067 



Tanned ox-hide upon cast iron... 0-365 



0-108 



0-100 
0-115 



0-646 
0-100 

"o-ibo 



0-100 



0-100 
0-100 



0-100 
0-103 



0-106 



0-108 



Unguents. 



Tallow. 

Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Hog's lard. 
Olive oil. 
Dry soap. 

J Greased and satu- 
\ rated with water. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Olive oil. 
/ Hog's lard and 
\ plumbago. 
Tallow. 
Water. 
Soap. 
Tallow. 
Hog's lard. 
Olive oil. 
/ Hog's lard and 
(^ plumbago. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Olive oil. 
Tallow. 

f Lard and plum- 
tbago. 
Olive oil. 
Olive oil. 
Tallow. 
Hog's lard. 
Olive oil. 
Tallow. 
Hog's lard. 
Tallow. 
Olive oil. 
f Lard and plum- 
t bago. 

J Greased and satu- 
t rated with water. 



The extent of the surfaces in these experiments bore such a relation to the pres- 
sure as to cause them to be separated from one another throughout by an inter- 
posed stratum of the unguent. 



STRENGTH OF MATERIALS. 



301 



Table of the Results of Experiments on the Friction of Gudgeons 
or Axle-ends, in motion upon their hearings. By M. Morin. 



Surfaces in Contact. 


state of the Surfaces. 


Coefficient of Friction. 




f Coated with oil of olives, ") 








with hog's lard, tallow, > 


0-07 to 0-08 


Cast iron axles in 
cast iron bearings. 


- 


and soft gome J 

With the same and water... 
Coated with asphaltum 


0-08 
0-054 






Greasy..... 


0-14 








0-14 






' Coated with oil of olives, \ 








with hog's lard, tallow, V 


0-07 to 0-08 


Cast iron axles in 


. 


and soft gome J 




cast iron bearings. 


Greasy 


0-16 






Greasy and damped 


0-16 






Scarcely greasy 


0-19 


Wrought iron axles 




r Coated with oil of olives, '\ 




in cast iron bear- 




tallow, hog's lard, or l 


0-07 to 0-08 


ings. 




soft gome 




Wrought iron axles 
in brass bearings. 




'' Coated with oil of olives, 1 
hog's lard, or tallow, / 

Coated with hard gome 

Greasy and wetted 


0-07 to 0-08 

0-09 
0-19 






Scarcely greasy 


0-25 


Iron axles in lignum 




r Coated with oil or hog's 1 

lard 1 

Greasy 


Oil 


vitae bearings. 




0-19 


Brass axles in brass 


r Coated with oil 


0-10 


bearings. 


t With hog's lard 


0-09 



Table of Coefficients of Friction under Pressures increased continu- 
ally up to limits of Abrasion. 







Coefficients of Friction. 












Square Inch. 


Wrought Iron upon 


Wrought Iron upon 


steel upon Cast 


Brass upon Cast 




"Wrought Iron. 


Cast Iron. 


Iron. 


Iron. 


32-5 lbs. 


•140 


•174 


•166 


•157 


l-66cwts. 


•250 


•275 


•300 


•225 


2-00 


•271 


•292 


•333 


•219 


2-33 


•285 


•321 


•340 


•214 


2-66 


•297 


•329 


•344 


•211 


3-00 


•312 


•333 


•347 


•215 


3-33 


•350 


•351 


•351 


•206 


3-66 


•376 


•353 


•353 


•205 


4-00 


•395 


•365 


•354 


•208 


4-33 


•403 


•366 


•356 


•221 


4-66 


•409 


•366 


•357 


•223 


5-00 




•367 


•858 


•233 


5-33 




•367 


•359 


•234 


5-66 




•367 


•367 


•235 


6-00 




•376 


•403 


•233 


6-33 




•434 




•234 


6-66 








•235 


7-00 








•232 


7-33 








•273 



2 A 



302 THE PRACTICAL MODEL CALCULATOR. 

Comparative friction of steam engines of different modifications, 
if the beam engine be taken as the standard of comparison : — 

The vibrating engine has a gain of 1*1 per cent. 

The direct-action engine, with slides — loss of 1-8 — 

Ditto, with rollers — gain of 0*8 — 

Ditto, with a parallel motion — gain of 1*3 — 

Excessive allowance for friction has hitherto been made in cal- 
culating the effective power of engines in general ; as it is found 
practically, by experiments, that, where the pressure upon the pis- 
ton is about 12 lbs. per square inch, the friction does not amount 
to more than IJ lbs. ; and also that, by experiments with an indi- 
cator on an engine of 50 horse power, the whole amount of friction 
did not exceed 5 horse power, or one-tenth of the whole power of 
the engine. 

RECENT EXPERIMENTS MADE BY M. MORIN ON THE STIFFNESS OF ROPES, 
OR THE RESISTANCE OF ROPES TO BENDING UPON A CIRCULAR ARC. 

The experiments upon which the rules and table following are 
founded were made by Coulomb, with an apparatus the invention 
of Amonton, and Coulomb himself deduced from them the follow- 
ing results : — 

1. That the resistance to bending could be represented by an 
expression consisting of two terms, the one constant for each rope 
and each roller, which we shall designate by the letter A, and 
which this philosopher named the natural stiffness, because it de- 
pends on the mode of fabrication of the rope, and the degree of 
tension of its yarns and strands ; the other, proportional to the 
tension, T, of the end of the rope which is being bent, and which 
is expressed by the product, BT, in which B is also a number 
constant for each rope and each roller. 

2. That the resistance to bending varied inversely as the diame- 
ter of the roller. 

Thus the complete resistance is represented by the expression 
A -f BT 
D ' 
where D represents the diameter of the roller. 

Coulomb supposed that for tarred ropes the stiffness was pro- 
portional to the number of yarns, and M. Navier inferred, from 
examination of Coulomb's experiments, that the coefficients A and 
B were proportional to a certain power of the diameter, which de- 
pended on the extent to which the cords were worn. M. Morin, 
however, deems this hypothesis inadmissible, and the following is 
an extract from his new work, "Logons de Mecanique Pratique," 
December, 1846 : — 

" To extend the results of the experiments of Coulomb to ropes 
of different diameters from those which had been experimented 
upon, M. Navier has allowed, very explicitly, what Coulomb had 
but surmised ; that the coefficients, A, were proportional to a cer- 



STRENGTH OF MATERIALS. 



303 



tain power of the diameter, which depended on the state of wear 
of the ropes ; but this supposition appears to us neither borne out, 
nor even admissible, for it would lead to this consequence, that a 
worn rope of a metre diameter would have the same stiffness as a 
new rope, which is evidently wrong ; and, besides, the comparison 
alone of the values of A and B shows that the power to which the 
diameter should be raised would not be the same for the two terms 
of the resistance." 

Since, then, the form proposed by M. Navier for the expression 
of the resistance of ropes to bending cannot be admitted, it is ne- 
cessary to search for another, and it appears natural to try if the 
factors A and B cannot be expressed for white ropes, simply accord- 
ing to the number of yarns in the ropes, as Coulomb has inferred 
for tarred ropes. 

Now, dividing the values of A, obtained for each rope by M. 
Navier, by the number of yarns, we find for 



n==SO d = 0'"-200 A 



0-222460 - 
n 



0-0074153. 



n=:l5d=- 0'^-144 A = 0-063514 - = 0-0042343. 

n 

n^Qd = 0^^-0088 A = 0-010604 - = 0-0017673. 

n 

It is seen from this that the number A is not simply propor- 
tional to the number of yarns. 

Comparing, then, the values of the ratio — corresponding to 
the three ropes, we find the following results : — 



Number of 
yarns. 


Values of 
A 

n ' 


Differences of the numbers of 
yarns. ' 


Differences of 

the values of 

A 


Differences of 
the values of 

— for eaoh 

yarn of 
difference. 


30 

15 

6 


0-0074153 
0-0042343 
0-0017673 


From 30 to 15. 15 yarns 

— 15 to 6. 9 — 

— 30 to 6. 24 — 


0-0031810 0-000212 
0-0024770 0-000272 
0-0056400 1 0-000252 



Mean difference per yarn, 0-000245 

It follows, from the above, that the values of A, given by the 
experiments, will be represented with sufficient exactness for all 
practical purposes by the formula 

A = 71 [0-0017673 -f 0-000245 {n — 6)1. 
= n [0-0002973 -f 0-000245 w]. 

An expression relating only to dry white ropes, such as were used 
by Coulomb in his experiments. 

With regard to the number B, it appears to be proportional to 
the number of yarns, for we find for 



304 THE PRACTICAL MODEL CALCULATOR. 

n = S0 d = 0^-0200 B = 0-009738 - = 0-0003246 

n 

T> 

n = l5d = 0°^-0144 B = 0-005518 - = 0-0003678 

n 

n = 6d = 0^^-0088 B = 0-002380 - = 0-0003967 

71 

Mean 0-0003630 

Whence 

B = 0-000363 n. 

Consequently, tlie results of the experiments of Coulomb on dry 
white ropes will be represented with sufficient exactness for prac- 
tical purposes by the formula 

K=n [0-000297 + 0-000245 n + 0-000363 T] kil. 
which will give the resistance to bending upon a drum of a metre 
in diameter, or by the formula 

R = ^ [0-000297 + 0-000245 n + 0-000363 T] kil. 

for a drum of diameter D metres. 

These formulas, transformed into the American scale of weights 
and measures, become 

U = n [0-0021508 + 0-0017724 n + 0-00119096 T] lbs. 
for a drum of a foot in diameter, and 

R = ^ [0-0021508 -f- 0-0017724 n + 0.00119096 T] lbs. 

for a drum of diameter D feet. 

With respect to worn ropes, the rule given by M. Navier cannot 
be admitted, as we have shown above, because it would give for the 
stiffness of a rope of a diameter equal to unity the same stiffness 
as for a new rope. 

The experiments of Coulomb on worn ropes not being sufficiently 
complete, and not furnishing any precise data, it is not possible, 
without new researches, to give a rule for calculating the stiffness 
of these ropes. 

TARRED ROPES. 

In reducing the results of the experiments of Coulomb on tarred 
ropes, as we have done for white ropes, we find the following 
values : — 

01 = 30 yarns A = 0-34982 B = 0-0125605 
n = U — A = 0-106003 B = 0-006037 
n= 6 — A = 0-0212012 B = 0-0025997 

which differ very slightly from those which M. Navier has given. 
But, if we look for the resistance corresponding to each yarn, we 
find 



STRENGTH OF MATERIALS. 



305 



n = SO yarns 
71 = 15 — 
n = 6 — 



- = 0-0116603 
n 

= 0-00T0662 



- = 0-000418683 
n 



n 

- = 0-0035335 
n 



= 0-000402466 



- = 0-000433283 
n 

Mean 0-000418144 



We see by this that the value of B is for tarred ropes, as for 
white ropes, sensibly proportional to the number of yarns, but it 
is not so for that of A, as M. Navier has supposed. 

Comparing, as we have done for white ropes, the values of — 

corresponding to the three ropes of 30, 15, and 6 yarns, we obtain 
the following results : — . 



Number of 
yarns. 


Values of 
A 
n 


Differences of the number of 
yarns. 


Differences of 

the values of 

A 


Differences of 
the values of 

-^ for each 

yarn of 

difference. 


30 
15 

6 


0-0116603 
0-0070662 
0-0035335 


From 30 to 15. 15 yarns 

— 15 to 6. 9 — 

— 60 to 6. 25 — 


0-0045941 
0-0035327 
0-0081268 


0-000306 
0-000392 
0-000339 



Mean 0-000346 

It follows from this that the value of A can be represented by 
the formula 

A = n [0-0035335 + 0-000346 (n - 6)] 
= n [0-0014575 + 0-000346 n^ 
and the whole resistance on a roller of diameter D metres, by 

71/ 

R = - [0-0014575 + 0-000346 n + 0-000418144 T] kil. 

Transforming this expression to the American scale of weights and 
measures, we have 

R = ^ [0-01054412 + 0-00250309 w + 0-001371889 T] lbs. 

for the resistance on a roller of diameter D feet. 

This expression is exactly of the same form as that which relates 
to white ropes, and shows that the stiffness of tarred ropes is a little 
greater than that of new white ropes. 

In the following table, the diameters corresponding to the differ- 
ent numbers of yarns are calculated from the data of Coulomb, by 

the formulas, 

d cent. = v/0-1338 n for dry white ropes, and 
d cent. = ■s/0.186 n for tarred ropes, 
which, reduced to the Americaii scale, become 

d inches = v/0'020739 n for dry white ropes, and 
d inches = v'0-02883 for tarred ropes. 
2a2 20 



306 



THE PRACTICAL MODEL CALCULATOR. 



Note. — The diameter of the rope is to be included in D ; thus, 
with an inch rope passing round a pulley, 8 inches in diameter in 
the groove, the diameter of the roller is to be considered as 9 
inches. 



I 

o 




Dry White Ropes. 






Tarred Ropes. 




Diameter. 


Value of the natural 
stiflfness, A. 


Value of the stiff- 
ness proportional 
to the tension, B. 


Diameter. 


Value of the natural 
stiffness, A. 


Value of the stiff- 
ness proportional 
to the tension, B. 




ft. 


lbs. 




ft. 


lbs. 




6 


00293 


0-0767120 


0-0071457 


0-0347 


0-153376 


0-00823133 


9 


0.0360 


0-1629234 


0-0107186 


00425 


0-297647 


001-234700 


1-2 


0-0416 


0-2810384 


0-014-2915 


, 0-0490 


0-486976 


001646267 


15 


004'35 


0-4310571 


0-0178644 


0-0548 


0-721357 


0-02057834 


18 


0-0509 


0-6129795 


00214373 


0-0600 


0-000795 


0-02469400 


21 


0-0550 


0-8268054 


0-0250102 


0-0648 


1-325289 


0-02880967 


24 


0-0588 


1.0725350 


0-0285831 : 


0-0693 


1-694839 


0-03292534 


27 


0-0622 


1-3501682 


003-21559 


0-0735 


2-109444 


0-0.3704100 


30 


0-0657 


1-6597051 


0-0357288 i 


0-0775 


2-569105 


0-041156»37 


33 


0-0689 


2-0011455 


0-0393017 


0-0S13 


3-073821 


0-04527234 


86 


0-0720 


2-3744897 


00428746 


0-0849 


3-623593 


0-04938800 


39 


0-0749 


2-7797375 


0-0464475 


6-0884 


4-218416 


0-05350367 


42 


0-0778 


3-21688S8 


0-0500203 


0-0917 


4-858304 


0-05761934 


45 


0-0805 


3-6859438 


0-0535932 


0-0949 


5-543242 


0-06173501 


48 


0-08.31 


4-1869024 


0-0571661 


0-0980 


6-273-237 


0-06585067 


51 


0-0857 


4-7197647 


0-0607390 


0-1010 


7-048287 


0-06996634 


54 


0-0882 


5-2845306 


0-0643119 j 


0-1040 


7-868393 


0-07408201 


57 


0-0908 


5-S812001 


0-0678847 


0-1070 


8-733554 


0-07819767 


60 


0-0926 


6-5097733 
r 0-0021508ra 


0-0714576 


0-1099 


964.3771 
r 0-01054412ra 


0-08-231334 


n 


>'0-000144re 


|+0-0017724reJ 


0-00119096?i 


yO-00020ra 


|+000250309n^ 


0-001371889n 



Application of the preceding Tables or Formulas. 

To find the stiffness of a rope of a given diameter or number of 
yarns, we must first obtain from the table, or by the formulas, the 
values of the quantities A and B corresponding to these given 
quantities, and knowing the tension, T, of the end to be wound 
up, we shall have its resistance to bending on a drum of a foot in 
diameter, by the formula 

R = A + BT. 

Then, dividing this quantity by the diameter of the roller or 
pulley round which the rope is actually to be bent, we shall have 
the resistance to bending on this roller. 

What is the stiffness of a dry white rope, in good condition, of 
60 yarns, or '0928 diameter, which passes over a pulley of 6 inches 
-diameter in the groove, under a tension of 1000 lbs. ? The table 
gives for a dry white rope of 60 yarns, in good condition, bent 
upon a drum of a foot in diameter, 

A = 0-50977 B = 0-0714576 
and we have D = 0*5 + 0-0928 ; and consequently, 



R = 



6-50977 + 0-0714576 x 1000 



= 128 lbs. 



0-5928 

The whole resistance to be overcome, not including the friction 
on the axis, is then 

Q 4- R = 1000 -\- 128 = 1128 lbs. 
The stiffness in this case augments the resistance by more than 
one-eighth of its value. 



STRENGTH OF MATERIALS. 307 

Further recent experiments made by m. morin, on the trac- 
tion OF CARRIAGES, AND THE DESTRUCTIVE EFFECTS WHICH THEY 
PRODUCE UPON THE ROADS. 

The study of the effects which are produced when a carriage is 
set in motion can be divided into two distinct parts : the traction 
of carriages, properly so called, and their action upon the roads. 

The researches relative to the traction of carriages have for their 
object to determine the magnitude of the effort that the motive 
power ought \o exercise according to the weight of the load, to the 
diameter and breadth of the wheels, to the velocity of the carriage, 
and to the state of repair and nature of the roads. 

The first experiments on the resistance that cylindrical bodies 
offer to being rolled on a level surface are due to Coulomb, who 
determined the resistance offered by rollers of lignum vitae and 
elm, on plane oak surfaces placed horizontally. 

His experiments showed that the resistance was directly propor- 
tional to the pressure, and inversely proportional to the diameter 
of the rollers.' 

If, then, P represent the pressure, and r the radius of the roller, 
the resistance to rolling, K,, could, according to the laws of Cou- 
lomb, be expressed by the formula 

R = A- 

r 

in which A would be a number, constant for each kind of ground, 
but varying with different kinds, and with the state of their 
surfaces. 

The results of experiments made at Vincennes show that the 
law of Coulomb is approximately correct, but that the resistance 
increases as the width of the parts in contact diminishes. 

Other experiments of the same nature have confirmed these con- 
clusions ; and we may allow, at least, as a law sufficiently exact 
for practical purposes, that for woods, plasters, leather, and gene- 
rally for hard bodies, the resistance to rolling is nearly — 

1st. Proportional to the pressure. 

2d. Inversely proportional to the diameter of the wheels. 

3d. Greater as the breadth of the zone in contact is smaller. 

EXPERIMENTS UPON CARRIAGES TRAVELLING ON ORDINARY ROADS. 

These experiments were not considered sufficient to authorize 
the extension of the foregoing conclusions to the motion of car- 
riages on ordinary roads. It was necessary to operate directly on 
the carriages themselves, and in the usual circumstances in which 
they are placed. Experiments on this subject were therefore un- 
dertaken, first at Metz, in 1837 and 1838, and afterwards at Cour- 
bevoie, in 1839 and 1841, with carriages of every species ; and 
attention was directed separately to the influence upon the magni- 
tude of the traction, of the pressure, of the diameter of the wheels, 
of their breadth, of the speed, and of the state of the ground. 

In heavily laden carriages, which it is most important to take 



308 



THE PRACTICAL MODEL CALCULATOR. 



into consideration, tlie weight of the wheels may be neglected in 
comparison with the total load ; and the relation between the load 
and the traction, upon a level road, is approximately given by the 
equation — 

E 2 f A X fr ) 

p^— / w v/ ^^^' carriages with four wheels, 

¥ A xfr^ 
and Tr= — :: ^ for carriages with two wheels. 



Pi r 

in which F^ represents the horizontal component of the traction ; 
Pj the total pressure on the ground ; 
/ and r" the radii of the fore and hind wheels ; 
r^ the mean radius of the boxes ; 
/ the coefficient of friction ; 
and A the constant multiplier in Coulomb's formula for the 
resistance to rolling. 

These expressions will serve us hereafter to determine, by aid of 
experiment, the ratio of the traction to the load for the most usual 
cases. 

Influence of the Pressure. 
To observe the influence of the pressure upon the resistance to 
rolling, the same carriages were made to pass with different loads 
over the same road in the same state. 

The results of some of these experiments, made at a walking pace, 
are given in the following table : — 











Ratio of the 


Carriages employed. 


Eoad trayersed. 


Pressure. 


Traction. 


traction to 
the load. 






kil. 


kil. 




Chariot porte corps 


Road from Courbe- 


6992 


180-71 


1/38-6 


d'artillerie. 


voie to Colomber, 


6140 


159-9 


1/39-2 




dry, in good re- 


4580 


113-7 


1/40-2 




pair, dusty. 








Chariot deroulage, 


Road from Courbe- 


7126 


138-9 


1/51-3 


without springs. 


voie to Bezous, 


5458 


115-5 


1/48-9 




solid, *hard gra- 


4450 


93-2 


1/47-7 




vel, very dry. 


3430 


68-4 


1/50-2 


Chariot de roulage, 


Road from Colomber 


1600 


39-3 


1/40-8 


with springs. 


to Courbevoie, 


3292 


89-2 


1/36-9 




pitched, in ordina- 


4996 


136-0 


1/36-8 




ry repair,! muddy 








Carriages with six 


Road from Courbe- 


3000 


138-9 


1/21-6 


equal wheels. 


voie to Colomber, 


4692 


224-0 


1/21-0 


Two carriages with 


deep ruts, with 


6000 


285 -,8 


1/21-0 


six equal wheels, 


muddy detritus. 


6000 


286-7 


1/21-0 


hooked on, one 










behind the other. 











From the examination of this table, it appears that on |solid 
gravel and on pitched roads the resistance of carriages to traction 
is sensibly proportional to the pressure. 

* En gravier dur. f Pav6 en etat ordinaire. J En empierrement solide. 



STRENGTH OF MATERIALS. 



309 



We remark that the experiments made upon one and upon two 
six- wheeled carriages have given the same traction for a load of 
6000 kilogrammes, including the vehicle, whether it was borne 
upon one carriage or upon two. It follows thence that the trac- 
tion is, cseteris paribus and between certain limits, independent of 
the number of wheels. 

Influence of the Diameter of the Wheels. 

To observe the influence of the diameter of the wheels on the 
traction, carriages loaded with the same weights, having wheels 
with tires of the same width, and of which the diameters only were 
varied between very extended limits, were made to traverse the 
same parts of roads in the same state. Some of the results obtained 
are given in the following table. 

These examples show that on solid roads it may be admitted as 
a practical law that the traction is inversely proportional to the 
diameters of the wheels. 







Diameter of 


Diameter of 








-i 












the wheels in 


the wheels in 


sT 


67" 


Eatioof 


-^ a 




Value 


Value 






metres. 


EngUsh feet. 


— u 


* 


the trac- 


= 2 


Resist- 


of A 


of A 


Carriages employed. 


Roads traversed. 






|s 


I 


tion to 


S = 


ance to 


for the 


for the 






















^s 


■S 


the pres- 


■|§ 


roUing, 


French 


American 






Fore 


Hind 


Fore 


Hind 


£ 


a 


sure. 


u 


R. 


scale. 


scale. 






wheels 


wheels 


wheels 


wheels 


p. 


a 














2 7-' 


2r" 


2r' 


2r" 








z 








1 


m. 


~^' 






kil. 


kil. 




kil. 


kil. 






Chariot porte Koad from Cour- 


2-029 


2-029 


6-657 


6-6.57 


4928 


81-6 


1/60- 


9-6 


72-0 10-0148 


0-01856 


corps d'artil- 


beA'oie to Colom- 


1-453 


1-453 


4-767 


4-767 


4930 


108-6 


1/45-5 


14-4 


94-2 1 0-0139 


0-04560 


lerie. 


ber, *solid gra- 
vel, dusty. 


0-872 


0-872 


2-861 


2861 


4924 


179-0 


1/27-4 


25-3 


153-7 jO-0137 


0-04494 


Porte corps d'ar- 


1 


2-029 


2-029 


6-657 


6-657 


4692 


51-45 


1/90-45 


9-0 


42-45 0-0092 


003018 


tillerie. 




1-453 


1-453 


4-767 


4-767 


4594 


71-45 


1/64-3 


13-2 


58-25 0-0092 


0-0301S 


Chariot comtois. 

A six-wheeled 

carriage. i 

The same with j 


fPitched pave- 

J- ment of Fon- 

tainebleau. 


1-110 

0-860 


1-358 
0-860 


3-642 
2-822 


4-455 
2-822 


1871 
3270 


32-10 

81-05 


1/58-4 
1/40-4 


4-7 
9-7 


27-40 
71-35 


0-0089 
0-0094 


0-02920 
0-03084 


four wheels. : 




0-860 


0-860! -2-822 


2-822 


3-270 


78-80 


1/41-5 


9-7 


69-10 0-0091 


0-02986 


Camion. , 




0-59-2 


0-66Uil-;j42l2-165 


1500 


52-30 


1/28-8 


8-8 


43-50! 0-0091 


0-02986 


Camion. 




0-420 


0'597{l-37Sll-y59 


1600 


68-20 


1/22-4 


11-6 


56-60 0-0089 

1 


0-02920 



Influence of the Width of the Felloes. 
Experiments made upon wheels of different breadths, having the 
same diameter, show, 1st, tkat on soft ground the resistance to 
rolling increases as the width of the felloe ; 2dly, on solid gravel 
and pitched roads, the resistance is very nearly independent of the 
width of the felloe. 

Influence of the Velocity. 

To investigate the influence of the velocity on the traction of 
carriages, the same carriages were made to traverse different roads 
in various conditions ; and in each series of experiments the velo- 
cities, while all other circumstances remained the same, underwent 
successive changes from a walk to a canter. 

Some of the results of these experiments are given in the follow- 
ing table : — 



* Empierrement solide. 



f Pav6 en grfes. 



310 



THE PRACTICAL MODEL CALCULATOR. 





j 




Rate of 




•Ratio 


Carriage employed. 


Ground passed over. 


Load. 


Pace. 


speed, 

in miles, 

per 

hour. 


Trac- 
tion. 


of the 

traction 

to the 

load. 






kil. 




miles. 


kil. 




Apparatus upon a 


Ground of the po- 


1042 


Walk 


3-13 


165-0 


1/6-32 


brass shaft. 


lygon at Metz, 
■wet and soft. 




Trot 


6-26 


168-0 


1/6-2^ 






1335 


Walk 


2-860 


215-0 


1/6-21 








Trot 


7-560 


197-0 


1/6-78 


A sixteen-pounder 


Road from Metz 


3750 


Walk 


2-820 


92- 


1/40-8 


carriage and 


to Montigny, 




*Brisk walk 


3-400 


92- 


1/40-8 


piece. 


solid gravel, 




Trot 


5-480 


102- 


1/36-8 




very even and 




fCanter.'.... 


8-450 


121- 


1/81- 




very dry. 












Chariot des Mes- 


Pitched road of 


3288 


Walk 


2-770 


144- 


1/22-8 


sageries, sus- 


Fontainebleau. 












pended upon six 




3353 


*Brisk walk 


3-82 


153- 


1/21-9 


springs. 






Trot 


5-28 


161- 


1/20-8 








JBrisk trot. 


8-05 


183-5 


1/18-3 



We see, by these examples, that the traction undergoes no sen- 
sible augmentation with the increase of velocity on soft grounds ; 
but that on solid and uneven roads it increases with an increase of 
velocity, and in a greater degree as the ground is more uneven, and 
the carriage has less spring. 

To find the relation between the resistance to rolling and the ve- 
locity, the velocities were set off as abscissas, and the values of A 
furnished by the experiments, as ordinates ; and the points thus 
determined were, for each series of experiments, situated very 
nearly upon a straight line. The value of A, then, can be repre- 
sented by the expression, 

A = a + c? (Y - 2) 
in which a is a number constant for each particular state of each 
kind of ground, and which expresses the value of the number A for 
the velocity, V = 2 miles, (per hour,) which is that of a very slow 
walk. 

d, a factor constant for each kind of ground and each sort of 
carriage. 

The results of experiments made with a carriage of a siege train, 
with its piece, gave, on the Montigny road, §very good solid gravel, — 
A = 0-03215 X 0-00295 (V - 2). 

On the lipitched road of Metz, A = 0-01936 x 0-08200 (Y - 2). 

These examples are sufficient to show — 

1st. That, at a walk, the resistance on a good pitched road is 
less than that on very good solid gravel, very dry. 

2d. That, at high speeds, the resistance on the pitched road in- 
creases very rapidly with the velocity. 

On rough roads the resistance increases with the velocity much 
more slowly, however, for carriages with springs. 



* Pas allonge. f Grand trot. 

I En tres bon empierrement. 



+ Trot allonge. 
Pav6 en grfes de Sieack; 



STRENGTH OF MATERIALS. 311 

Thus, for a chariot des Messageries G^nerales, on a pitched road, 
the experiments gave A = 0-0117 X 0-00361 (V — 2) ; while, with 
the springs wedged so as to prevent their action, the experiments 
gave, for the same carriage, on a similar road, A = 0'02T23 x 
0*01312 (V — 2). At a speed of nine miles per hour, the springs 
diminish the resistance by one-half. 

The experiments further showed that, while the pitched road was 
inferior to a *solid gravel road when dry and in good repair, the 
latter lost its superiority when muddy or out of repair. 

INFLUENCE OF THE INCLINATION OF THE TRACES. 

The inclination of the traces, to produce the maximum effect, is 
given by the expression — 

Ax 0-96/ r^ 
^f - r - 0-4: fr' 
in which h = the height of the fore extremity of the trace above 
the point where it is attached to the carriage ; b = the horizontal 
distance between these two points. / is the radius of interior of 
the boxes, and r the radius of the wheel. 

The inclination given by this expression for ordinary carriages 
is very small ; and for trucks with wheels of small diameter it is 
much less than the construction generally permits. 

It follows, from the preceding remarks, that it is advantageous 
to employ, for all carriages, wheels of as large a diameter as can 
be used, without interfering with the other essentials to the pur- 
poses to which they are to be adapted. Carts have, in this respect, 
the advantage over wagons ; but, on the other hand, on rough roads, 
the thill horse, jerked about by the shafts, is soon fatigued. Now, 
by bringing the hind wheels as far forward as possible, and placing 
the load nearly over them, the wagon is, in effect, transformed into 
a cart ; only carp must be taken to place the centre of gravity of 
the load so far in front of the hind wheels that the wagon may not 
turn over in going up hill. 

ON THE DESTRUCTIVE EFFECTS PRODUCED BY CARRIAGES ON THE ROADS. 

If we take stones of mean diameter from 2f to 3J inches, and, 
on a road slightly moist and soft, place them first under the small 
wheels of a diligence, and then under the large wheels, we find that, 
in the former case, the stones, pushed forward by the small wheels, 
penetrate the surface, ploughing and tearing it up ; while in the 
latter, being merely pressed and leant upon by the large wheels, 
they undergo no displacement. 

From this simple experiment we are enabled to conclude that 
the wear of the roads by the wheels of carriages is greater the 
smaller the diameter of the wheels. 

Experiments having proved that on hard grounds the traction 
was but slightly increased when the breadths of the wheels was 

* En empierrement. 



312 THE PRACTICAL MODEL CALCULATOR. 

diminislied, we might also conclude that the wear of the road would 
be but slightly increased by diminishing the width of the felloes. 

Lastly, the resistance to rolling increasing with the velocity, it 
was natural to think that carriages going at a trot would do more 
injury to the roads than those going at a walk. But springs, by 
diminishing the intensity of the impacts, are able to compensate, 
in certain proportions, for the effects of the velocity. 

Experiments, made upon a grand ^ale, and having for their 
object to observe directly the destructive effects of carriages upon 
the roads, have confirmed these conclusions. 

These experiments showed that with equal loads, on a solid gra- 
vel road, wheels of two inches breadth produced considerably more 
wear than those of 4J inches, but that beyond the latter width there 
was scarcely any advantage, so far as the preservation of the road 
was concerned, in increasing the size of the tire of the wheel. 

Experiments made with wheels of the same breadth, and of dia- 
meters of 2-86 ft., 4-77 ft., and 6-69 ft., showed that after the 
carriage of 10018-2 tons, over tracks 218*72 yards long, the track 
passed over by the carriage with the smallest wheels was by far 
the most worn ; while, on that passed over by the carriage with 
the wheels of 6'69 ft. diameter, the wear was scarcely perceptible. 

Experiments made upon two wagons exactly similar in all other 
respects, but one with and one without springs, showed that the 
wear of the roads, as well as the increase of traction, after the 
passage of 4577'36 tons over the same track, was sensibly the same 
for the carriage without springs, going at a walk of from 2*237 to 
2*684 miles per hour, and for that with springs, going at a trot of 
from 7*158 to 8*053 miles per hour. 



HYDRAULICS. 

THE DISCHARGE OF WATER BY SIMPLE ORIFICES AND TUBES. 

The formulas for finding the quantities of water discharged in a 
given time are of an extensive and complicated nature. The more 
important and practical results are given in the following Deduc- 
tions. 

When an aperture is made in the bottom or side of a vessel con- 
taining water or other homogeneous fluid, the whole of the particles 
of fluid in the vessel will descend in lines nearly vertical, until they 
arrive within three or four inches of the place of discharge, when 
they will acquire a direction more or less oblique, and flow directly 
towards the orifice. 

The particles, however, that are immediately over the orifice, de- 
scend vertically through the whole distance, while those nearer to 
the sides of the vessel, diverted into a direction more or less oblique 
as they approach the orifice, move with a less velocity than the 
former ; and thus it is that there is produced a contraction in the 
size of the stream immediately beyond the opening, designated the 
vena contracta, and bearing a proportion to that of the orifice of 



HYDI^AULICS. 313 

about 5 to 8, if it pass through a thin plate, or of 6 to 8, if through 
a short cylindrical tube. But if the tube be conical to a length 
equal to half its larger diameter, having the issuing diameter less 
than the entering diameter in the proportion of 26 to 33, the stream 
does not become contracted. 

If the vessel be kept constantly full, there will flow from the 
aperture twice the quantity that the vessel is capable of contain- 
ing, in the same time in whi^h it would have emptied itself if not 
kept supplied. 

1. How many horse-power (H. P.) is required to raise 6000 
cubic feet of water the hour from a depth of 300 feet ? 

A cubic foot of w^ater weighs 62*5 lbs. avoirdupois. 

6000 X 62-5 

WK = 6250, the weight of water raised a minute. 

6250 X 300 = 1875000, the units of work each minute. 

^, 1875000 

Then oqqqa = 56-818 = the horse-power required. 

2. What quantity of water may be discharged through a cylin- 
drical mouth-piece 2 inches in diameter, under a head of 25 feet ? 

2 1 
Jo = "^ of a foot; .*. the area of the cross section of the 

mouth-piece, in feet, is ^ x -^ x -7854 = -021816. 



Theory gives -021816 V'2 ^ X 25 the cubic feet discharged each 
second ; but experiments show that the effective discharge is 97 per 
cent, of this theoretical quantity: g = 32*2. 

Hence, -97 X -021816 n/64-4 x 25 = -84912, the cubic feet 
discharged each second. 

•84912 X 62-5 = 53-0688 lbs. of water discharged each second. 

Effluent water produces, by its vis viva, about 6 per cent, less me- 
chanical effect than does its weight by falling from the height of 
the head. 

3. What quantity of water flows through a circular orifice in a 
thin horizontal plate, 3 inches in diameter, under a head of 49 feet ? 

Taking the contraction of the fluid vein into account, the velo- 
city of the discharge is about 97 per cent, of that given by theory. 

The theoretic velocity isy2g X 49 = 7 v/6'44 = 56-21. 
-97 X 56-21 = 54-523 = the velocity of the discharge. 
The area of the transverse section of the contracted vein is -64 
of the transverse section of the orifice. 

3 1 

— = - = -25, and (-25)2 x -7854 = -0490875 = area of orifice. 

.-. -64 X -0490875 = -031416, the area of the transverse section 
of the contracted vein. 
2B 



314 THE PRACTICAL MODEL CALCULATOR. 

Hence, 54-523 x -031416 = 1-7129, the cubic feet of water 
discharged each second. The later experiments of Poncelet, 
Bidone, and Lesbros give '563 for the coefficient of contraction. 
Water issuing throug,h lesser orifices give greater coefficients of 
contraction, and become greater for elongated rectangles, than for 
those which approach the form of a square. 

Observations show that the result above obtained is too great ; 
^ of this result are found to be very near the truth. 

^ of 1-7129 = 1-0541. 

4. What quantity of water flows through a rectangular aperture 
7*87 inches broad, and 3*94 inches deep, the surface of the water 
being 5 feet above the upper edge ; the plate through which the 
water flows being -125 of an inch thick. 

7-87 

-3-^ = -65583, decimal of a foot. 

3-94 

—^ = -32833, decimal of a foot. 

5- and 5-32833 are the heads of water above the uppermost and 
lowest horizontal surfaces. 

The theoretical discharge will be 

g X -65583 ^/2y ((5-328)^ - (5)^) = 3-9268 cubic feet. 

Table I. gives the coefficient of efflux in this case, -615, which 
is found opposite 5 feet and under 4 inches ; for 3-94 is nearly 
equal 4. 

3-9268 X -615 = 2-415 cubic feet, the effective discharge. 

5. What water is discharged through a rectangular orifice in a 
thin plate 6 inches broad, 3 inches deep, under a head of 9 feet 
measured directly over the orifice ? 

6 
jq = -5, decimal of a foot. 

3 

tq = '25, decimal of a foot. 

The theoretical discharge will be 

3 X -5 ^/% I (9-25f - (9f I == 3-033 cubic feet. 

Table 11. gives the coefficient of efflux between -604 and -606 ; 
we shall take it at -605, then 

3-033 X -605 = 1-833 cubic feet, the effective discharge. 

6. A weir -82 feet broad, and 4-92 feet head of water, how many 
cubic feet are discharged each second ? 

The quantity will be 

c X -82 x/2^ (4-92)3; g = 32-2; 



HYDRAULICS. 



315 



Table I. — The Coefficients foi^ the Efflux through rectangular ori- 
fices in a thin vertical plate. The heads are measured where the 
ivater may be considered still. 



Head of water, or 














distance of the 












surface of the 














water from the 
upper side of 
the orifice in 
feet. 














In. 

8- 


In. 

4; 


In. 

2- 


In. 
1- 


In. 

•8 


In. 

•4 


•1 


■579 


•599 


•619 


•634 


•656 


•686 


•2 


•582 


•601 


•620 


•638 


•654 


•681 


•3 


•585 


•603 


•621 


•640 


•653 


•676 


•4 


•588 


•605 


•622 


•639 


•652 


•671 


•5 


•591 


•607 


•623 . 


•637 


•650 


•666 


•6 


•594 


•609 


•624 


•635 


•649 


•662 


•7 


•596 


•611 


•625 


•634 


•648 


•659 


•8 


•597 


•613 


•623 


•632 


•647 


•656 


•9 


•598 


•615 


•627 


•631 


•645 


•653 


1-0 


•599 


•616 


•628 


•630 


•644 


•650 


2-0 


•600 


•617 


•628^ 

•62er 


•628 


•641 


•647 


3-0 


•601 


•617 


•626 


•638 


•644 


4-0 


•602 


•616 


•624 


•623 


•634 


•640 


5-0 


•604 


•615 


•621 


•621 


•630 


•635 


6-0 


•608 


•613 


•618 


•618 


•625 


•630 


7-0 


•002 


•611 


•615 


•615 


•621 


•625 


8-0 


•601 


•609 


•612 


•613 


•617 


•619 


9-0 


•600 


•606 


•609 


•610 


•614 


•613 


10-0 


•600 


•604 


•606 


•608 


•611 


•609 



Table II. — The Coefficients for the Efflux through rectangular ori- 
fices in a thin vertical plate, the heads of water being measured 
directly over the orifice. 



Head of water, or 
distance of the 
surface of the 
water from the 
upper side of 
the orifice in 
feet. 






EIGHT or 


Orifice. 






In. 
8^ 


In. 

4- 


In. 

2' 


In. 
1^ 


In. 

•8 


In. 

4 


•1 


•593 


•613 


•637 


•659 


•685 


•708 


•2 


•593 


•612 


•636 


•656 


•680 


•701 


•3 


•593 


•613 


•635 


•653 


•676 


•694 


•4 


•594 


•614 


•634 


•650 


•672 


•687 


•5 


•595 


•614 


•633 


•647 


•668 


•681 


•6 


•597 


•615 


•632 


•644 


•664 


•675 


•7 


•598 


•615 


•631 


•641 


•660 


•669 


•8 


•599 


•616 


•630 


•638 


•655 


•663 


•9 


•601 


•616 


•629 


•635 


•650 


•657 


10 


•603 


.•617 


•629 


•632 


•644 


•651 


2^0 


•604 


•617 


•626 


•628 


•640 


•646 


3^0 


•605 


•616 


•622 


•627 


•636 


•641 


4^0 


•604 


•614 


•618 


•624 


•632 


•636 


5-0 


•604 


•613 


•616 


•621 


•628 


•631 


60 


•603 


•612 


•613 


•618 


•624 


•626 


7^0 


•603 


•610 


•611 


•616 


•620 


•621 


8-0 


•602 


•608 


•609 


•614 


•616 


•617 


9^0 


•601 


•607 


•607 


•612 


•613 


•613 


10^0 


•601 


•603 


•606 


•610 


•610 


•609 



316 



THE Pira.CTICAL MODEL CALCULATOR. 



c is termed the coefficient of efflux, and on an average may be taken 
at -4. It is found to vary from -385 to -444. 

X -82 ^/(64•4) (4-92p 



2-670033, the cubic feet dis- 



Then -4 
charged each second. 

7. What breadth must be given to a notch, in a thin plate, with 
a head of water of 9 inches, to allow 10 cubic feet to flow each 
second ? 

The breadth will be represented by 

10 10 

- • = ^ = 4-7963 feet. 

c s/lg X (-75)^ ^ -4 X x/64-4 x (-75)^ 

Changes in the coefficients of efflux through convergent sides 
often present themselves in practice : they occur in dams which are 
inclined to the horizon. 

Poncelet found the coefficient -8, when the board was inclined 
45°, and the coefficient -74 for an inclination of 63° 34', that is 
for a slope of 1 for a base, anS 2 for a perpendicular. 

8. If a sluice board, inclined at an angle of 50°, which goes 
across a channel 2-25 feet broad, is drawn out -5 feet, what quan- 
tity of water will be discharged, the surface of the water standing 
4- feet above the surface of the channel, and the coefficient of efflux 
taken at -78 ? 

The height of the aperture = -5 sin. 50° = -3830222 ; 4- and 
4- - -3830222 = 3-6169778, are the heads of water. 

.-. |x 2-25 X -78 X v/% |(4f - (3-617)H = 10-5257 cu- 
bic feet, the quantity discharged. 

The calculations just made appertain to those cases where the 
water flows from all sides towards the aperture, and forms a co*q- 
tracted vein on every side. We shall next calculate in cases where 
the water flows from one or more sides to the aperture, and hence 
produces a stream only a 
partially contracted. 



n, 0, p. 



are four orifices in 




the bottom ABCD of a 
vessel ; the contraction by 
efflux through the orifice 
0, in the middle of the bot- 
tom, is general, as the water 
can flow to it from all 
sides ; the contraction c 
from the efflux through m, 7i, p, is partial, as the water can only 
flow to them from one, two, or three sides. Partial contraction 
gives an oblique direction to the stream, and increases the quantity 
discharged. 

9. What quantity of water is delivered through a flow 4 feet 
broad, and 1 foot deep, vertical aperture, at a pressure of 2 feet 
above the upper edge, supposing the lower edge to coincide with 



HYDRAULICS. 



817 



the lower side of the channel, so that there is no contraction at the 
bottom ? 

The theoretical discharge will be 

^ X J X V2g I (3f - (2f I = o0'668 cubic feet. ' 

The coefficient of contraction given in the table page 315, may 
be taken at '603. 

I. — Qomparison of the Theoretical withthe Real Discharges fi^om an 

Orifice. 



Constant height 

of the water in the 

reservoir above 

the centre of the 

orifice. 


Theoretical dis- 
charge through a 
circular orifice 
one inch in di- 
ameter. 


Eeal discharge 

in the same time 

through the same 

orifice. 


Ratio of the 

theoretical to the real 

discharge. 


Paris Feet. 
1 


Cvibic Inches. 

4881 


Cubic Inches. 

2722 


1 to 0-62133 


2 


6196 


S846 


1 to 0-62073 


3 


7589 


4710 


1 to 0-62064 


4 


8763 


5436 


1 to 0-62034 


5 


9797 


6075 


1 to 0-62010 


6 


10732 


6654 


1 to 0-62000 


7 


11592 


7183 


1 to 0-61965 


8 


12392 


7672 


1 to 0-61911 


9 


13144 


8135 


1 to 0-61892 


10 


13855 


8574 


1 to 0-61883 


11 


14530 


8990 


1 to 0-61873 


12 


15180 


9384 


1 to 0-61819 


13 


15797 


9764 


1 to 0-61810 


14 


16393 


10130 


1 to 0-61795 


15 


16968 


10472 


1 to 0-61716 



II. — Com'parison of the Theoretical with the Real Discharges from 

a Tube. 



Constant height 

of the water in the 

reservoir above 

the centre of the 

orifice. 


Theoretical dis- 
charge through a 
circular orifice 
one inch in di- 
ameter. 


Real discharge 
in the same time 

by a cylindrical 

tube one inch in 

diameter and two 

inches long. 


Ratio of the 

theoretical to the real 

discharge. 


Paris Feet. 
1 


Cubic Inches. 

4381 


Cubic Inches. 

3539 


1 to 0-81781 


2 


6196 


5002 


1 to 0-80729 


3 


7589 


6126 


1 to 0-80724 


4 


8763 


7070 


1 to 0-80681 


5 


9797 


7900 


1 to 0-80638 


6 


10732 


8654 


1 to 0-80638 


7 


11592 


9340 


1 to 0-80577 


8 


12392 


9975 


1 to 0-80496 


9 


13144 


10579 


1 to 0-80485 


10 


13855 


11151 


1 to 0-80483 


11 


14530 


11693 


1 to 0-80477 


12 


15180 


12205 


1 to 0-80403 


13 


15797 


12699 


1 to 0-80390 


14 


16393 


13177 


1 to 0-80382 


15 


16968 


13620 


1 to 0-80270 



2 B 2 



318 THE PRACTICAL MODEL CALCULATOR. 

THE DISCHARGE BY DIFFERENT APERTURES AND TUBES, UNDER DIF- 
FERENT HEADS OF WATER. 

The velocity of water flowing out of a horizontal a2je7^ture, is as 
the square root of the height of the head of the ivater. — That is, the 
pressure, and consequently the height, is as the square of the ve- 
locity ; for, the quantity flowing out in any short time is as the 
velocity ; and the force required to produce a velocity in a certain 
quantity of matter in a given time is also as that velocity; there- 
■fore, the force must be as the square of the velocity. 

Or, supposing a very small cylindrical plate of water, imme- 
diately over the orifice, to be put in motion at each instant, by the 
pressure of the whole cylinder upon it, employed only in generat- 
ing its velocity ; this plate w6uld be urged by a force as much 
greater than its own weight as the column is higher than itself, 
through a space shorter in the same proportion than that height. 
But where the forces are inversely as the spaces described, the 
final velocities are equal. Therefore, the velocity of the water 
flowing out must be equal to that of a heavy body falling from the 
height of the head of water ; which is found, very nearly, by mul- 
tiplying the square root of that height in feet by 8, for the number 
of feet described in a second. Thus, a head of 1 foot gives 8 ; a 
head of 9 feet, 24. This is the theoretical velocity ; but, in con- 
sequence of the contraction of the stream, we must, in order to ob- 
tain the actual velocity, multiply the square root of the height, in 
feet, by 5 instead of 8. 

The velocity of a fluid issuing from an aperture is not afi'ected 
by its density being greater or less. Mercury and water issue 
with equal velocities at equal altitudes. 

The proportion of the theoretical to the actual velocity of a fluid 
issuing through an opening in a thin substance, according to M. 
Eytelwein, is as 1 to '619 ; but more recent experiments make it 
as 1 to -621 up to -645. 

APPLICATION OP THE TABLES IN THE PRECEDING PAGE. 

Table I. — To find the quantities of ivater discharged hy orifices 
of different sizes under different altitudes of the fluid in the reser- 
voir. 

To find the quantity of fluid discharged by a circular aperture 
3 inches in diameter, the constant altitude being 30 feet. 

As the real discharges are in the compound ratio of the area of 
the apertures and the square roots of the altitudes of the water, 
and as the theoretical quantity of water discharged by an orifice 
one inch in diameter from a height of 15 feet is, by the second co- 
lumn of the table, 16968 cubic inches in a minute, we have this 
proportion : 1 v/15 : 9 n/30 : : 16968 : 215961 cubic inches ; the 
theoretical quantity required. This quantity being diminished in 
the ratio of 1 to '62, being the ratio of the theoretical to the ac- 
tual discharge, according to the fourth column of the table, gives 
133896 cubic inches for the actual quantity of water discharged by 



HYDRAULICS. 319 

the given aperture. Hence, the quantity should be rather greater, 
because large orifices discharge more in proportion than small ones ; 
while it should be rather less, because the altitude of the fluid 
being greater than that in the table ^yith which it is compared, the 
flowing vein of water becomes rather more contracted. The quan- 
tity thus found, therefore, is nearly accurate as an average. 

When the orifice and altitude are less than those in the table, a 
few cubic inches should be deducted from the result thus derived. 

The altitude of the fluid being multiplied by the coefficient 8*016 
will give its theoretical velocity ; and as the velocities are as the 
quantities discharged, the real velocity may be deducted from the 
theoretical by means of the foregoing results. 

Table II. — To find the quantities of water discharged hy tubes 
of different diameter, and under different heights of water. 

To find the quantity of water discharged by a 'cylindrical tube, 
4 inches in diameter, and 8 inches long, the constant altitude of 
the water in the reservoir being 25 feet. 

Find, in the same manner as by the example to Table L, the 
theoretical quantity discharged, which is furnished by this analogy. 
1 v/15 : 16 v^25 : : 16968 : 350490 cubic inches, the theoretical 
discharge. This, diminished in the ratio of 1 to '81 by the 4th 
column, will give 28473 cubic inches for the actual quantity dis- 
charged. If the tube be shorter than twice its diameter, the 
quantity discharged will be diminished, and approximate to that 
from a simple orifice, as shown by the production of the vena con- 
tracta already described. 

According to Eytelwein,«the proportion of the theoretical to the 
real discharge through tubes, is as follows : 

Through the shortest tube that will cause the stream to adhere 
everywhere to its sides, as 1 to 0-8125. 

Through short tubes, having their lengths from two to four 
times their diameters, as 1 to 0*82. 

Through a tube projecting within the reservoir, as 1 to 0*50. 

It should, however, be stated, that in the contraction of the 
stream the ratio is not constant. It undergoes perceptible varia- 
tions by altering the form and position of the orifice, the thickness 
of the plate, the form of the vessel, and the velocity of the issu- 
ing fluid. 

Deductions from experiments made hy Bossut, 3Iichelloti. 

1. That the quantities of fluid discharged in equal times from 
difi'erent-sized apertures, the altitude of the fluid in the reser- 
voir being the same, are to each other nearly as the area of the aper- 
tures. 

2. That the quantities of water discharged in equal times by 
the same orifice under diff'erent heads of water, are* nearly as the 
square roots of the corresponding heights of water in the reservoir 
above the centre of the apertures. 



320 THE PRACTICAL MODEL CALCULATOR. 

3. That, in general, the quantities of water discharged, in the 
same time, by different apertm-es under different heights of -water 
in the reservoir, are to one another in the compound ratio of the 
areas of the apertures, and the square roots of the altitudes of the 
water in the reservoirs. 

4. That on account of the friction, the smallest orifice discharges 
proportionally less water than those which are larger and of a. 
simikir figure, under the same heads of water. 

5. That, from the same cause, of several orifices whose areas 
are equal, that w*hich has the smallest perimeter will discharge 
more water than the other, under the same altitudes of water in 
the reservoir. Hence, circular apertures are must advantageous, as 
they have less rubbing surface under the same area. 

t). That, in consequence of a slight augmentation which the 
contraction of the fluid vein undergoes, in proportion as the height 
of the fluid in the reservoir increases, the expenditure ought to .be 
a little diminished. 

7. That the discharge of a fluid through a cylindrical horizontal 
tube, the diameter and length of which are equal to one another, 
is the same as through a simple orifice. 

8. That if the cylindrical horizontal tube be of greater length 
than the extent of the diameter, the discharge of water is much 
increased. 

9. That the length of the cylindrical horizontal tube may be 
increased with advantage to four times the diameter of the orifice. 

10. That the diameters of the apertures and altitudes of water 
in the reservoir being the same, the theoretic discharge through a 
tkin aperture, which is supposed to have no contraction in the vein, 
the discharge through an additional cylindrical tube of greater 
length than the extent of its diameter, and the actual discharge 
through an aperture pierced in a thin substance, are to each other 
as the numbers 16, 18, 10. 

11. That the discharges by different additional cylindrical tubes, 
under the same head of water, are nearly proportional to the areas 
of the orifices, or to the squares of the diameters of the orifices. 

12. That the discharges by additional cylindrical tubes of the 
same diameter, under different heads of water, are nearly propor- 
tional to the square roots of the head of water. 

13. That from the two preceding corollaries it follows, in gene- 
ral, that the discharge during the same time, by different addi- 
tional tubes, and under different heads of water in the reservoir, 
are to one another nearly in the compound ratio of the squares 
of the diameters of the tubes, and the square roots of the heads 
of water. 

The discharge of fluids by additional tubes of a conical figure, 
when the inner to the outer diameter of the orifice is as 33 to 
26, is augmafited very nearly one-seventeenth and seven-tenths 
more than by cylindrical tubes, if the enlargement be not carried 
too far. 



HYDRAULICS. 321 

DISCHARGE BY COMPOUND TUBES. 

Deductions from the experiments of M. Venturi. 

In the discharge by compound tubes, if the part of the addi- 
tional tube nearest the reservoir have the form of the contracted 
vein, the expenditure will be the same as if the fluid were not con- 
tracted at all ; and if to the smallest diameter of this cone a cylin- 
drical pipe be attached, of the same diameter as the least section 
of the contracted vein, the discharge of the fluid will, in a horizon- 
tal direction, be lessened by the friction of the water against the 
side of the pipe ; but if the same tube be applied in a vertical 
direction, the expenditure will be augmented, on the principle of 
the gravitation of falling bodies; consequently, the greater the 
length of pipe, the more abundant is the discharge of fluid. 

If the additional compound tube have a cone applied to the op- 
posite extremity of the pipe, the expenditure will, under the same 
head of water, be increased, in comparison with that through a 
simple orifice, in the ratio of 24 to 10. 

In order to produce this singular eff"ect, the cone nearest to the 
reservoir must be of the form of the contracted vein, which will 
increase the expenditure in the ratio of 12'1 to 10. At the other 
extremity of the pipe, a truncated conical tube must be applied, 
of which the length must be nearly nine times the smaller diameter, 
and its outward diameter must be 1*8 times the smaller one. This 
additional cone will increase the discharge in the proportion of 
24 to 10. But if a great length of pipe intervene, this additional 
tube has little or no efi"ect on the quantity discharged. 

According to M. Venturi's experiments on the discharge of 
water by bent tubes, it appears that while, with a height of water 
in the reservoir of 32*5 inches, 4 Paris cubic feet were discharged 
through a cylindrical horizontal tube in the space of 45 seconds, 
the discharge of the same quantity through a tube of the same 
diameter, with a curved end, occupied 50 seconds, and through a 
like tube bent at right angles, 70 seconds. Therefore, in making 
cocks or pipes for the discharge or conveyance of water, great 
attention should be paid to the nature and angle of the bendings ; 
right angles should be studiously avoided. 

The interruption of the discharge by various enlargements of 
the diameter of the tubes having been investigated by M. Venturi, 
by means of a tube with a diameter of 9 lines, enlarged in several 
parts to a diameter of 24 lines, the retardation was found to in- 
crease nearly in proportion to the number of enlargements ; the 
motion of the fluid, in passing into the enlarged parts, being 
diverted from its direct course into eddies against the sides of the 
enlargements. From which it may be deduced, that if the inter- 
nal roughness of a pipe diminish the expenditure, the friction of 
the water against these asperities does not form any considerable 
part of the cause. A right-lined tube may have its internal sur- 
face highly polished throughout its whole length, and it may every- 

21 



322 



THE PRACTICAL MODEL CALCULATOR. 



v.-here possess a diameter greater than the orifice to which it is 
applied; but, nevertheless, the expenditure will be greatly retarded 
if the pipe should have enlarged parts or swellings. It is not 
enough that elbows and contractions be avoided ; for it maj hap- 
pen, by an intermediate enlargement, that the whole of the other 
advantage may be lost. This will be obvious from the results in 
the following table, deduced from experiments with tubes having 
various enlargements of diaftieter. 



Head of water 
in inches. 


Number of en- 
larged parts. 


Seconds in which 

4 cubic feet were 

discharged. 


32-5 
32-5 
32-5 
32-5 




1 

3 

5 


109 
147 
192 
240 



DISCHARGE BY CONDUIT PIPES. 

On account of the friction against the sides, the less the dia- 
meter of the pipe, the less proportionally is the discharge of fluid. 
And, from the same cause, the greater the length of conduit pipe, 
the greater the diminution of the discharge. Hence, the dis- 
charges made in equal times by horizontal pipes of different lengths, 
but of the same diameter, and under the same altitude of water, 
are to one another in the inverse ratio of the square roots of the 
lengths. In order to have a perceptible and continuous discharge 
of fluid, the altitude of the water in the reservoir, above the axis 
of the conduit pipe, must not be less than If inch for every 180 
feet of the length of the pipe. 

The ratio of the difference of discharge in pipes, 16 and 24 lines 
diameter respectively, may be known by comparing the ratios of 
Table I. with the ratios of Table II., in the following page. 

The greater the angle of inclination of a conduit pipe, the 
greater will be the discharge in a given time ; but when the angle 
of the conduit pipe is 6° 31', or the depression of the lower extre- 
mity of the pipe is one-eighth or one-ninth of its length, the rela- 
tive gravity of the fluid will be counterbalanced by the resistance 
or friction against the sides ; and the discharge is then the same 
as by an additional horizontal tube of the same diameter. 

A curvilinear pipe, the altitude of the water in the reservoir being 
the same, discharges less water when the flexures lie horizontally, 
than a rectilinear pipe of the same diameter and length. 

The discharge by a curvilinear pipe of the same diameter and 
length, and under the same head of w\ater, is still further dimi- 
nished when the flexures lie in a vertical instead of a horizontal plane. 

When there is a number of contrary flexures in a large pipe, the 
air sometimes lodges in the highest parts of the flexures, and greatly 
retards the motion of the water, unless prevented by air-holes, or 
stopcocks. 



HYDRAULICS. 



323 



Table I. — Comparison of the discharge hy conduit pipes of different 
lengths, 16 lines in diameter^ with the discharge hy additional 
tubes inserted in the same reservoir. — By M. BossuT. 



Constant 

altitude of the 

Water above the 

centre of the 

aperture. 


Length of 

the conduit 

ripe. 


Quantity of Water discharged 
in a minute. 


Ratio between the 

quantities furnished 

by tube and pipe. 


bv additional 

tube, 16 lines in 

diameter. 


by conduit 

pipe, 16 lines in 

diameter. 


Feet. 


Feet. 


Cubic Inches. 


Cubic Inches. 






30 


6330 


2778 


100 to 43-39 




60 


6330 


1957 


100 to 30-91 




90 


6330 


1587 


100 to 25-07 




120 


6330 


1351 


100 to 21-34 




150 


6330 


1178 


100 to 18-61 




180. 


6330 


1052 


100 to 16-62 


2 


30 


8939 


4066 


100 to 45-48 


2 


60 


8939 


2888 


100 to 32-31 


2 


90 


8939 


2352 


100 to 26-31 


2 


120 


8939 


2011 


100 to 22-50 


2 


150 


8939 


1762 


100 to 19-71 


2 


180 


8939 


1583 


100 to 17-70 



Table II. — Comparison of the discharge hy conduit pipes of dif- 
ferent lengths, 24 lines in diameter, with the discharge hy addi- 
tional tubes i7iserted in the same reservoir. — By M. BossuT. 



Constant 

altitude of the 

Water above the 

centre of the 

aperture. 


Length of 

the conduit 

pipe. 


Quantity of Water discharged 
in a minute. 


Ratio between the 

quantities furnished 

by tube and pipe. 


by additional 
tube, 24 lines in 


by conduit 
pipe, 21 lines in 




diameter. 


diameter. 




Feet. 


Feet. 


Cubic Inches. 


Cubic Inches. 




1 


30 


14243 


7680 


100 to 53-92 


1 • 


60 


14243 


5564 


100 to 39-06 


1 


90 


14243 


4534 


100 to 31-83 


1 


120 


14243 


3944 


100 to 27-69 


1 


150 


14243 


3486 


100 to 24-48 


1 


180 


14243 


3119 


100 to 21-90 


2 


30 


20112 


11219 


100 to 55-78 


2 


60 


20112 


8190 


100 to 40-72 


2 


90 


20112 


6812 


100 to 33-87 


2 


120 


20112 


5885 


100 to 29-26 


2 


150 


20112 


5232 


100 to 26-01 


2 


180 


20112 


4710 


100 to 23-41 



DISCHARGE BY WEIRS AND RECTANGULAR APERTURES. 

Rectangular orifices in the side of a reservoir, extending to the surface. 

The velocity varying nearly as the square root of the height, 
may here be represented by the ordinates of a parabola, and the 
quantity of water discharged by the area of the parabola, or 
two-thirds of that of the circumscribing rectangle. So that the 
quantity discharged may be found by taking two-thirds of the velo- 
city due to the mean height, and allowing for the contraction of 
the stream, according to the form of the opening. 

In a lake, for example, in the side of which a rectangular open- 
ing is made without any oblique lateral walls, three feet wide, and 



324 THE PRACTICAL MODEL CALCULATOR. 

extending two feet below the surface of the water, the coefficient 
of the velocity, corrected for contraction, is 5*1, and the corrected 
mean velocity f •■v/2 X 5*1 = 4*8 ; therefore the area being 6, the 
discharge of water in a second is 28*8 cubic feet, or nearly four 
hogsheads. 

The same coefficient serves for determining the discharge over 
a weir of considerable breadth ; and, hence, to deduce the depth 
or breadth requisite for the discharge of a given quantity of water. 
For example, a lake has a weir three feet in breadth, and the sur- 
face of the water stands at the height of five feet above it : it is 
required how much the weir must be widened, in order that the 
water may be a foot lower. Here the velocity is | v/5 X 5*1, and the 
quantity of water f -v/o X 5*1 X 3 x 5 ; but the velocity must be re- 

-T r . -, 1 , . .., , f n/5'x5-1 x3 x5 

duced to f \/4 X o-l, and then the section will be 



_ f v/4 X 5-1 

■v/5 X 3 X 5 

_ _ _ 'j'.5 X v/5 ; and the height being 4, the breadth 

7*5 — 
must be -T- n/5 = 4-19 feet. 

The discharge from reservoirs, with lateral orifices of consider- 
able magnitude, and a constant head of water, may be found by 
determining the difierence in the discharge by two open orifices of 
different heights ; or, in most cases, with nearly equal accuracy, 
by considering the velocity due to the distance, below the surface, 
of the centre of gravity of the orifice. 

Under the same height of water in the reservoir, the same quan- 
tity always flows in a canal, of whatever length and declivity ; but 
in a tube, a difference in length and declivity has a great effect on 
the quantity of water discharged. 

The velocity of water flowing in a river or stream varies at dif- 
ferent parts of the same transverse section. It is found to be 
greatest where the water is deepest, at somewhat less than one- 
half the depth from the surface ; diminishing towards the sides 
and shallow parts. 

Resistance to bodies moving in fluids. — The deductions from the 
experiments of C. Colles, (who first planned the Croton Aqueduct, 
New York,) and others, on this intricate subject, are, as stated, thus : 

1. The confirmation of the theory, that the resistance of fluids 
to passing bodies is as the squares of the velocities. 

2. That, contrary to the received opinion, a cone will move 
through the water with much less resistance with its apex foremost, 
than with its base forward. 

3. That the increasing the length of a solid, of almost any form, 
by the addition of a cylinder in the middle, diminishes the resist- 
ance with which it moves, provided the weight in the water remains 
the same. 



HYDRAULICS. 



325 



4. That the greatest breadth of the moving body should be 
placed at the distance of two-fifths of the whole length from the 
bow, when applied to the ordinary forms in naval architecture. 

5. That the bottom of a floating solid should be made triangu- 
lar ; as in that case it will meet with the least resistance when 
moving in the direction of its longest axis, and with the greatest 
resistance when moving with its broadside foremost. 

Friction of fluids. — Some experiments have been made on this 
subject, with reference to the motion of bodies in water, upon a 
cylindrical model, 30 inches in length, 26 inches in diameter, and 
weighing 255 lbs. avoirdupois. The cylinder was placed in a cis- 
tern of salt water, and made to vibrate on knife-edges passing 
through its axis, and was deflected over to various angles by means 
of a weight attached to the arm of a lever .^ The experiments were 
then repeated without the water, and the following are the angles 
of deflection and vibration in the two cases. 



In the salt water. 


In the 


itmosphere. 


Angle of 


Angle to -R-hicli 


Angle of 


Angle to -which 


Detection. 


it vibrated. 


Deflection. 


it vibrated. 


22° 30' 


22° 24' 


22° 30' 


20° 0' 


22 10 


22 6 


21 36 


21 3 


21 54 


21 48 


20 48 


2016 


21 36 


21 30 


&c. 


&c. 


&c. 


&c. 







Showing that the amplitude of vibration when oscillating in water 
is considerably less than when oscillating without water. In the 
experiments there is a falling off in the angle of 24', or nearly 
half a degree. The amount of force acting on the surface of the 
cylinder necessary to cause the above difi"erence was calculated ; 
and the author thinks that it is not equally distributed on the 
surface of the cylinder, but that the a.mount on any particular 
part might vary as the depth. On this supposition, a constant 
pressure at a unit of depth is assumed, and this, multiplied by the 
depth of any other point of the cylinder immersed in the water, 
will give the pressure at that point. These forces or moments 
being summed by integration and equated with the sum of the 
moments given by the experiments, we have the value of the con- 
stant pressure at a unit of depth = '0000469. This constant, in 
another experiment, the weight of the model being 197 lbs. avoir- 
dupois, and consequently the part immersed in the water being dif- 
ferent from that in the other experiment, was -0000452, which 
difliers very little from the former,—- indicating the probability of 
the correctness of the assumption. 

The drainage of water through pipes. — The experiments made 

under the direction of the Metropolitan Commissioners of Sewers, 

on the capacities of pipes for the drainage of towns, have presented 

some useful results for the guidance of those who have to make 

2C 



326 



THE PRACTICAL MODEL CALCULATOR. 



calculations for a similar purpose. The pipes, of yarious dia- 
meters, from 3 to 12 inches, were laid on a platform of 100 feet 
in length, the declivity of which could be varied from a horizontal 
level to a fall of 1 in 10. The water was admitted at the head 
of the pipe, and at five junctions, or tributary pipes on each side, 
so regulated as to keep the main pipe full. 

The results were as follow : — 

It was found — to mention only one result — that a line of 6-incli 
pipes, 100 feet long, at an inclination of 1 in 60, discharged 75 cubic 
feet per minute. The same experiment, repeated with the line of 
pipes reduced to 50 feet in length, gave very nearly the same result. 
Without the addition of junctions, the transverse sectional area of 
the stream of water near the discharging end was reduced to one- 
fifth of the corresponding area of the pipe, and it required a sim- 
ple head of water of about 22 inches to give the same result as 
that accruing under the circumstances of the junctions. With 
regard to varying sizes and inclinations, it appears, sufficiently for 
practical purposes, that the squares of the discharges are as the 
fifth powers of the diameters ; and again, that in steeper declivi- 
ties than 1 in 70, the discharges are as the square roots of the 
inclinations ; but at less declivities than 1 in 70, the ratios of the 
discharges diminish very rapidly, and are governed by no constant 
law. At a certain small declivity, the relative discharge is as the 
fifth root of the inclination ; at a smaller declivity, it is found as 
the seventh root of the inclination ; and so on, as it approaches the 
horizontal plane. This may be exemplified by the following results 
found by actual experiment : 

Discharges of a 6-inch pipe at several inclinations. 



Inclination. 


Discharges in lOU 
feet per minute. 


Inclination. 


Discharges in 100 
feet per minute. 


lin 60 


75 


lin 320 


49 


lin 80 


68 


lin 400 


48-5 


1 in 100 


63 


lin 480 


48 


1 in 120 


59 


lin 640 


47-5 


1 in 160 


54 


lin 800 


47-2 


1 in 200 


52 


1 in 1200 


46-7 


1 in 240 


50 


Level 


46 



The conclusion arrived at is, that the requisite sizes of drains 
and sewers can be determined (near enough for practical purposes, 
as an important circumstance has to be considered in providing for 
the deposition of solid matter, which disadvantageously alters the 
form of the aqueduct, and contracts the water-way) by taking the 
result of the 6-inch pipe, under the circumstances before mentioned 
as a datum, and assuming that the squares of the discharges are 
as the fifth powers of the diameters. 

That at greater declivities than 1 in 70, the discharges are as 
the square roots of the inclinations. 



WATER ^YflEELS. 327 

That at less declivities than 1 in 70, the usual law will not 
obtain ; but near approximations to the truth may be obtained by 
observing the relative discharges of a pipe laid at various small 
inclinations. 

That increasing the number of junctions, at intervals, accele- 
rates the velocity of the main stream in a ratio which increases as 
the square root of the inclination, and which is greater than the 
ratio of resistance due to a proportionable increase in the length 
of the aqueduct. The velocity at which the lateral streams enter 
the main line, is a most important circumstance governing the flow 
of water. In practice, these velocities are constantly variable, 
considered individually, and always different considered collectively, 
so that their united effect it is difficult to estimate. Again, the 
same sewer at different periods may be quite filled, but discharges 
in a given time very different quantities of water. It should be 
mentioned that in the case of the 6-inch pipe, which discharged 
75 cubic feet per minute, the lateral streams had a velocity of 
a few feet per second, and the junctions were placed at an angle 
of about 35° with the main line. It is needless to say that all 
junctions should be made as nearly parallel with the main line as 
possible, otherwise the forces of the lateral currents may impede 
rather than maintain or accelerate the main streams. 



WATEE WHEELS. 

THE UNDERSHOT WHEEL. 



The ratio between the power and effect of an undershot wheel 
is as 10 to 3*18 ; consequently 31*43 lbs. of water must be expended 
per second to produce a mechanical effect equal to that of the esti- 
mated labour of an active man. 

The velocity of the periphery of the undershot wheel should be 
equal to half the velocity of the stream ; the float-boards should be 
so constructed as to rise perpendicularly from the water ; not more 
than one-half should ever be below the surface ; and from 3 to 5 
should be immersed at once, according to the magnitude of the 
wheel. 

The following maxims have been deduced from experiments : — 

1. The virtual or effective head of water being the same, the 
effect will be nearly as the quantity expended ; that is, if a mill, 
driven by a fall of water, whose virtual head is 10 feet, and which 
discharges 30 cubic feet of w^ater in a second, grind four bolls of 
corn in an hour ; another mill having the same virtual head, but 
which discharges 60 cubic feet of water, will grind eight bolls of 
corn in an hour. 

2. The expense of water being the same, the effect will be nearly 
as the height of the virtual or effective head. 

3. The quantity of water expended being the same, the effect is 
nearly as the square of its velocity ; that is, if a mill, driven by a 



328 THE PRACTICAL MODEL CALCULATOR. 

certain quantity of water, moving with the velocity of four feet per 
second, grind three bolls of corn in an hour ; another mill, driven 
by the same quantity of water, moving with the velocity of five 
feet per second, will grind nearly 4^^ bolls in the hour, because 
3 : 4^0 : : 42 : 5^ nearly. 

4. The aperture being the same, the effect will be nearly as the 
cube of the velocity of the water ; that is, if a mill driven by water, 
moving through a certain aperture, with the velocity of four feet 
per second, grind three bolls of corn in an hour ; another mill, 
driven by water, moving through the same aperture with the velo- 
city of five feet per second, will grind 5|| bolls nearly in an hour ; 
for as 3 : 5|f : : 4^ : 5^ nearly. 

The height of the virtual head of water may be easily deter- 
mined from the velocity of the water, for the heights are as the 
squares of the velocities, and, consequently, the velocities are as 
the square roots of the height. 

To calculate the proportions of undershot ivheels. — Find the per- 
pendicular height of the fall of water above the bottom of the mill- 
course, and having diminished this number by one-half the depth of 
the water where it meets the wheel, call that the height of the fall. 

Multiply the height of the fall, so found, by 64*348, and take the 
square root of the product, which will be the velocity of the water. 

Take one-half of the velocity of the water, and it will be the 
velocity to be given to the float-boards, or the number of feet they 
must move through in a second, to produce a maximum effect. 
Divide the circumference of the wheel by the velocity of its float- 
boards per second, and the quotient will be the number of seconds 
in which the w^heel revolves. Divide 60 by the quotient thus found, 
and the new quotient will be the number of revolutions made by 
the wheel in a minute. 

Divide 90, the number of revolutions which a millstone, 5 feet 
in diameter, should make in a minute, by the number of revolutions 
made by the wheel in a minute, the quotient will be the number of 
turns the millstone ought to make for one turn of the wheel. 
Then, as the number of revolutions of the wheel in a minute is to 
the number of revolutions of the millstone in a minute, so must 
the number of staves in the trundle be to the number of teeth in 
the wheel, (the nearest in whole numbers.) Multiply the number 
of revolutions made by the w^heel in a minute, by the number of 
revolutions made by the millstone for one turn of the wheel, and 
the product will be the number of revolutions made by the millstone 
in a minute. 

The effect of the water wheel is a maximum, when its circum- 
ference moves with one-half, or, more accurately, with three- 
sevenths of the velocity of the stream. 

THE BREAST WHEEL. 

The effect. of a breast wheel is equal to the effect of an under 
shot wheel, whose head of water is equal to the diflerence of level 



WATER WHEELS. 



329 



between the surface of water in the reservoir, and the part where 
it strikes the wheel, added to that of an overshot, whose height is 
equal to the difference of level between the part where it strikes 
the wheel and the level of the tail water. 

When the fall of water is between 4 and 10 feet, a breast 
wheel should be erected, provided there be enough of water ; an 
undershot should be used when the fall is below 4 feet, and an 
overshot wheel when the fall exceeds 10 feet. Also, when the fall 
exceeds 10 feet, it should be divided into two, and two breast wheels 
be erected upon it. 

Table for breast wheels. 





2 


5-g 


llfl 
ill! 

m 

K ^ g "= 


2 

Hi 

> 


5 1 


iH 


k 
if 

o ^ 


to 












H-t 








Feet. 


Feet. 


Feet. 


Feet. 


Sec. 




lbs. avr. 


Cubic ft. 


1 


0-17 


198-6 


0-75 


2-18 


1-92 


4-80 


1536 


74-30 


2 


0-34 


35-1 


1-50 


3-09 


2-72 


6-80 


1084 


37-15 


3 


0-51 


12-7 


2-26 


3-78 


3-33 


8-32 


886 


24-77 


4 


0-69 


6-2 


3-01 


4-36 


3-84 


9-60 


762 


18-57 


5 


0-86 


3-57 


3-76 


4-88 


4-28 


10-70 


680 


14-86 ^ 


6 


1-03 


2-25 


4-51 


5-35 


4-70 


11-76 


626 


12-38 


7 


1-20 


1-53 


6-26 


5-77 


5-08 


12-70 


581 


10-61 


8 


1-37 


MO 


6-02 


6-17 


5-43 


13-58 


543 


9-29 


9 


1-54 


0-81 


6-77 


6-55 


5-76 


14-40 


512 


8-26 


10 


1-71 


0-77 


7-52 


6-90 


6-07 


15-18 


486 


7-43 



It is evident, from the preceding table, that when the height of 
the fall is less than 3 feet, the depth of the float-boards is so great, 
and their breadth so small, that the breast wheel cannot well be 
employed ; and, on the contrary, when the height of the fall ap- 
proaches to 10 feet, the depth of the float-boards is too small in 
proportion to their breadth ; these two extremes, therefore, must 
be avoided in practice. The ninth column contains the quantity 
of water necessary for impelling the wheel ; but the total expense 
of water should always exceed this by the quantity, at least, which 
escapes between the mill-course and the sides and extremities of 
the float-boards. 

THE OVERSHOT WHEEL. 

The ratio between the power and effect of an overshot wheel, is 
as 10 to 6'6, when the water is delivered above the apex of the 
wheel, and is computed from the whole height of the fall ; and as 
10 to 8 when computed from the height of the wheel only ; con- 
sequently, the quantity of water expended per second, to produce 
a mechanical effect equal to that of the aforesaid estimated labour 
of an active man, is, in the first instance, 15*15 lbs., and in the 
second instance, 12-5 lbs. 

Hence, the effect of the overshot wheel, under the same circum- 
2 c2 



330 



THE PRACTICAL MODEL CALCULATOR. 



stances of quantity and fall, is, at a medium, double that of the 
undershot. 

The velocity of the periphery of an overshot wheel should be 
from 6^ to 8J feet per second. 

The higher the wheel is, in proportion to the whole descent, the 
greater will be the effect. 

And from the equality of the ratio between the power and effect, 
subsisting where the constructions are similar, we must infer that 
the effects, as well as the powers, are as the quantities of water and 
perpendicular heights multiplied together respectively. 

Working machinery by hyd.raulic pressure. — The vertical pressure 
of water, acting on la piston, for raising weights and driving machi- 
nery, is coming into use in many places where it can be advantage- 
ously applied. At Liverpool, Newcastle, Glasgow, and other places, 
it is applied to the working of cranes, drawing coal-wagons, and other 
purposes requiring continuous power. The presence of a natural fall, 
like that of Golway, Ireland, which can be conducted to the engine 
through pipes, is, of course, the most economical situation for the 
application of such power ; in other situations, artificial power must 
be used to raise the water, which, even under this disadvantage, may, 
from its readiness and simplicity of action, be often serviceably em- 
ployed. Wherever the contiguity of a steam engine would be dan- 
gerous, or otherwise objectionable, a water engine would afford the 
means of receiving and applying the power from any required dis- 
tance, precautions being taken against the action of frost on the fluid. 

Required the horse power of a centre discharging Tu7'hine water 
wheel, the head of water being 25 feet, and the area of the open- 
ing 400 inches. 

The following table shows the working horse power of both the 
inward and outward discharging Turbine water wheels ; they are 
calculated to the square inch of opening. 



Centr 


e Discharging 


Outward Discharg- 


Centre Discharging 


Outward Discharg- 




Turbine. 


ing Turbine. 




rurbine. 


ing Turbine. 


Head. 


Horse Power. 


Horse Power. 


Head. 


Horse Power. 


Horse Power. 


3 


•00821 


•012611 


22 


•19523 


•339972 


4 


•01483 


•025145 


23 


•20787 


•364182 


5 


•02137 


•038124 


24 


•22315 


•384615 


6 


•02685 


•045618 


25 


•23667 


•412013 


7 


•03414 


•058314 


26 


•25125 


•437519 


8 


•04198 


•074413 


27 


•26482 


•455698 


9 


•05206 


•089025 


28 


•28135 


•484427 


10 


•05883 


•106215 


29 


•29563 


•510833 


11 


•06921 


•118127 


30 


•30817 


•537721 


12 


•07851 


•135610 


31 


•32316 


•561425 


13 


•08882 


•150638 


32 


•33617 


•587148 


14 


•10054 


•173158 


33 


•34823 


•611013 


15 


•11002 


•192234 


34 


•36154 


•638174 


16 


•12093 


•211592 


35 


•37123 


•665164 


17 


•13196 


•231161 


36 


•39874 


•692156 


18 


•14275 


•257145 


37 


•40118 


•726148 


19 


•15613 


•273325 


38 


•41762 


•764115 


20 


•16927 


•296618 


39 


•42156 


•804479 


21 


•18109 


•317167 


40 


•43718 


•849814 



WATER WHEELS. 



331 



Opposite 25 in the column marked "Head," the working horse 
power to the square inch is found to be -25667, which, multiplied 
by 400, gives 94*668, the horse power required. 

What is the working horse power of an outward discharging 
Turbine, under the efiective head of 20 feet ; the area of all the 
openings being 325 square inches. In the table, opposite 20, we 
find -296618, then -296618 X 325 = 96-4, the required horse power. 

What is the number of revolutions a minute of an outward 
discharging Turbine wheel, the head being 19 feet and the dia- 
meter of the wheel 60 inches ? 

In the table for the outward discharging wheel, opposite 19, and 
under 60 inches, we find 97, the number of revolutions required. 

What is the number of revolutions a minute of an inward dis- 
charging Turbine, under a head of 21 feet, the diameter being 
72 inches ? 

In the table for the inward discharging wheel, opposite 21 feet, 
and under 72 inches, we find 95, the number of revolutions a 
minute. 

These Turbine tables were calculated by the author's brother, 
the late John O'Byrne, C. E., who died in Ncav York, on the 6th 
of April, 1851. 









Outward discharging 


Turbine, 










ll 




Diameter 


IN IXCHE3. 














24 


30 


36 


42 


48 


54 


60 


66 


72 


78 


84 


90 

27 


96 
21 


3 


100 


80 


70 


60 


52 


42 


37 


35 


32 


30 


28 


4 


111 


89 


73 


63 


57 


49 


44 


41 


37 


34 


32 


30 


28 


5 


123 


100 


82 


71 


62 


55 


51 


45 


42 


38 


37 


33 


31 


6 


135 


109 


91 


78 


68 


62 


55 


50 


45 


42 


38 


37 


36 


7 


146 


118 


96 


84 


73 


65 


59 


63 


49 


47 


42 


40 


38 


8 


156 


125 


105 


90 


79 


71 


63 


57 


52 


49 


43 


42 


39 


9 


166 


133 


111 


95 


83 


75 


67 


61 


57 


50 


49 


45 


41 


10 


175 


140 


117 


100 


87 


79 


70 


64 


59 


55 


51 


47 


46 


11 


183 


147 


122 


105 


92 


81 


74 


67 


62 


57 


54 


49 


48 


12 


191 


156 


127 


110 


96 


85 


79 


70 


64 


59 


55 


53 


51 


13 


200 


159 


133 


115 


100 


89 


81 


73 


67 


62 


57 


55 


53 


14 


206 


166 


138 


118 


104 


92 


83 


75 


69 


64 


59 


57 


55 


15 


213 


171 


142 


122 


107 


95 


86 


78 


72 


66 


61 


58 


56 


16 


222 


177 


148 


126 


111 


98 


89 


82 


74 


69 


64 


59 


57 


17 


227 


182 


152 


131 


115 


101 


91 


83 


77 


71 


66 


62 


59 


18 


234 


187 


156 


134 


117 


105 


94 


85 


78 


73 


67 


63 


61 


19 


238 


193 


161 


138 


120 


107 


97 


88 


81 


74 


69 


64 


63 


20 


247 


197 


164 


141 


124 


110 


99 


90 


84 


76 


71 


66 


64 


21 


252 


202 


168 


145 


126 


114 


101 


92 


85 


78 


73 


68 


65 


22 


259 


208 


172 


149 


129 


115 


105 


94 


87 


80 


74 


69 


67 


23 


263 


212 


176 


151 


133 


119 


106 


96 


89 


84 


77 


72 


70 


24 


270 


216 


180 


155 


135 


120 


109 


98 


92 


85 


78 


74 


72 


25 


277 


222 


184 


158 


138 


123 


111 


101 


93 


86 


80 


76 


74 


26 


282 


226^ 


189 


161 


141 


125 


113 


103 


95 


87 


81 


78 


76 


27 


286 


229 


191 


165 


143 


129 


116 


105 


97 


88 


83 


79 


77 


28 


291 


233 


195 


167 


146 


130 


118 


107 


99 


91 


85 


80 


78 


29 


297 


237 


199 


170 


149 


132 


119 


109 


100 


92 


80 


81 


80 


30 


303 


241 


202 


174 


152 


135 


122 


111 


102 


94 


88 


82 


81 



332 



THE PRACTICAL MODEL CALCULATOR. 









Inwm 


'd discharging 


Turhine. 










3 










Diameter 


IX Inches. 












24 
111 


30 


36 


42 


48 


54 


60 


66 


72 


78 


Oi 


90 


96 


86 


74 


62 


54 


48 


47 


40 


36 


32 


31 


30 


27 


4 


125 


96 


83 


70 


62 


55 


51 


45 


41 


37 


36 


34 


31 


5 


141 


112 


94 


78 


69 


61 


55 


50 


46 


43 


40 


37 


36 


6 


152 


122 


101 


86 


76 


67 


62 


55 


51 


47 


43 


42 


38 


7 


166 


131 


108 


93 


82 


72 


65 


60 


54 


51 


47 


44 


42 


8 


175 


139 


116 


99 


87 


76 


71 


.63 


57 


54 


49 


47 


45 


9 


186 


149 


123 


i06 


93 


81 


74 


68 


63 


57 


53 


51 


47 


10 


195 


156 


129 


111 


99 


86 


78 


71 


66 


61 


56 


52 


49 


11 


208 


167 


136 


117 


1©2 


91 


82 


74 


68 


63 


58 


56 


52 


12 


217 


169 


142 


122 


107 


97 


85 


78 


71 


66 


61 


57 


54 


13 


221 


178 


148 


127 


112 


99 


89 


82 


74 


69 


64 


61 


56 


14 


231 


184 


153 


133 


116 


104 


92 


85 


76 


71 


66 


62 


58 


15 


238 


191 


159 


136 


119 


107 


95 


87 


80 


73 


68 


64 


61 


16 


245 


198 


165 


144 


123 


111 


99 


90 


83 


76 


71 


66 


63 


17 


252 


203 


168 


148 


127 


114 


102 


92 


85 


78 


73 


68 


64 


18 


269 


209 


173 


150 


132 


116 


104 


95 


87 


82 


75 


69 


66 


19 


267 


215 


176 


153 


134 


120 


108 


98 


89 


83 


77 


72 


67 


20 


276 


222 


183 


157 


138 


122 


111 


101 


93 


85 


79 


74 


69 


21 


288 


226 


186 


162 


141 


125 


113 


103 


95 


86 


80 


75 


71 


22 


290 


230 


192 


164 


145 


129 


116 


107 


96 


89 


83 


77 


73 


23 


299 


235 


196 


167 


146 


133 


118 


109 


97 


91 


84 


79 


74 


24 


303 


240 


201 


171 


151 


135 


122 


111 


101 


93 


86 


80 


75 


25 


310 


247 


206 


176 


155 


138 


123 


112 


104 


96 


88 


82 


76- 


26 


314 


248 


210 


180 


157 


139 


126 


115 


106 


97 


90 


84 


79 


27 


319 


254 


213 


183 


162 


142 


128 


117 


108 


99' 


92 


85 


80 


28 


327 


261 


218 


186 


164 


146 


129 


119 


109 


102 


93 


87 


82 


29 


333 


265 


221 


189 


166 


148 


133 


121 


111 


103 


95 


89 


83 


30 


336 


271 


224 


193 


168 


151 


136 


124 


114 


105 


97 


90 


85 



WINDMILLS. 

1. The velocity of windmill sails, whether unloaded or loaded, 
so as to produce a maximum eflfect, is nearly as the velocity of the 
wind, their shape and position being the same. 

2. The load at the maximum is nearly, but somewhat less than, 
as the square of the velocity of the wind, the shape and position 
of the sails being the same. 

3. The effects of the same sails, at a maximum, are nearly, but 
somewhat less than, as the cubes of the velocity of the wind. 

4. The load of the same sails, at the maximum, is nearly as the 
squares, and their effect as the cubes of their number of turns in a 
given time. 

5. When sails are loaded so as to produce a maximum at a given 
velocity, and the velocity of the wind increases, the load continu- 
ing the same, — 1st, the increase of effect, when the increase of the 
velocity of the wind is small, will be nearly as the squares of those 
velocities ; 2dly, when the velocity of the wind is double, the ef- 
fects will be nearly as 10 to 27|- ; but, 3dly, when the velocities 
compared are more than double of that when the given load pro- 
duces a maximum, the effects increase nearly in the simple ratio 
of the velocity of the wind. 



WINDMILLS. 



333 



6. In sails where the figure and position are similar, and tlie ve- 
locity of the wind the same, the number of turns, in a given time, 
will be reciprocally as the radius or length of the sail. 

7. The load, at a maximum, which sails of a similar figure and 
position will overcome, at a given distance from the centre of mo- 
tion, will be as the cube of the radius. 

8. The efi'ects of sails of similar figure and position are as the 
square of the radius. 

9. The velocity of the extremities of Dutch sails, as well as of the 
enlarged sails, in all their usual positions when unloaded, or even 
loaded to a maximum, is considerably greater than that of the wind. 

The results in Table 1 are for Dutch sails, in their common posi- 
tion, when the radius was 30 feet. Table 2 contains the most 
efficient angles. 





1. 






2. 




Number of 
revolutions of 
wind-shaft in 

a minute. 


Velocity of 

the wind in 

an hour. 


Ratio between 
velocity of 

wind and re- 
volutions of 
wind-shaft. 


Parts of the 

radius, which 

is divided into 

six parts. 


Angle with 
the axis. 


Angle of weather. 


3 
5 
6 


2 miles 

4 miles 

5 miles 


0-666 
0-800 
0-833 


\ 
3 
4 
5 
6 


72° 
71 
72 
74 

77J 
83 


18° 

19 

18 middle 

16 



Supposing the radius of the sail to be 30 feet, then the sail will 
commence at ^, or 5 feet from the axis, where the angle of inclina- 
tion will be 72 degrees ; at |, or 10 feet from the axis, the angle 
will be 71 degrees, and so on. 

Results of ExiJeriments on the effect of Windmill Sails in grind- 
ing corn. — By M. Coulomb. 

A windmill, with four sails, measuring 72 feet from the ex- 
tremity of one sail to that of the opposite one, and 6 feet 7 inches 
wide, or a little more, was found capable of raising 1100 lbs. avoir- 
dupois 238 feet in a minute, and of working, on an average, eight 
hours in a day. This is equivalent to the work of 34 men, 30 square 
feet of canvas performing about the daily work of a man. 

When a vertical windmill is employed to grind corn, the mill- 
stone makes 5 revolutions in the same time that the sails and the 
arbor make 1. • 

The mill does not begin to turn till the velocity of the wind is 
about 13 feet per second. 

When the velocity of the wind is 19 feet per second, the sails 
make from 11 to 12 turns in a minute, and the mill will grind from 
880 to 990 lbs. avoirdupois in an hour, or about 22,000 lbs. in 24 
hours. 



834 



THE APPLICATION OF LOGARITHMS. 



The practice of performing calculations by Logarithms is an ex- 
ercise so useful to computers, that it requires a more particular ex- 
planation than could have been properly given in that part of the 
"W'ork allotted to Arithmetic. 

A few of the various applications, of logarithms, best suited to 
the calculations of the engineer and mechanic, have therefore been 
collected, and are, with other matter, given, in hopes that they will 
come into general use, as the certainty and accuracy of their re- 
sults can be more safely relied upon and more easily obtained 
than with common arithmetic. 

By a slight examination, the student will perceive, in some de- 
gree, the nature and effect of these calculations ; and, by frequent 
exercise, will obtain a dexterity of operation in every case admitting 
of their use. He will also more readily penetrate the plans of the 
different devices employed in instrumental calculations, which are 
rendered obscure and perplexing to most practical men by their ig- 
norance of the proper application of logarithms. . 

Logarithms are artificial numbers which stand for natural num- 
bers, and are so contrived, that if the logarithm of one number be 
added to the logarithm of another, the sum will be the logarithm 
of the product of these numbers ; and if the logarithm of one num- 
ber be taken from the logarithm of another, the remainder is the 
logarithm of the latter divided by the former ; and also, if the loga- 
rithm of a number be multiplied by 2, 3, 4; or 5, &c., we shall have 
the logarithm of the square, cube, &;c., of that number ; and, on the 
other hand, if divided by 2, 3, 4, or 5^ &c., we have the logarithm 
of the square root, cube root, fourth root, &c., of the proposed num- 
ber ; so that with the aid of logarithms, multiplication and division 
are performed by addition and subtraction ; and the raising of 
powers and extracting of roots are effected by multiplying or di- 
viding by the indices of the powers and roots. 

In the table at the end of this work, are given the logarithms of 
the natural numbers, from 1* to 1000000 by the help of differences ; 
in large tables, only the decimal part of the logarithm is given, as 
the index is readily determined ; for the index of the logarithm of 
any number greater than unity, is equal to one less than the num- 
ber of figures on the left hand of the decimal point ; thus, 

The index of 12345- is 4-, 

1234-5 - 3-, 

123-45 - 2-, 

12-345 - 1-, 

1-2345 - 0- 



THE APPLICATION OF LOGARITHMS. 335 

The index of any decimal fraction is a negative number equal to 
one and the number of zeros immediately following the decimal 
point; thus, _ 

The index of -00012345 is -4- or 4- 

-0012345 is -3- or 3- 

-012345 is -2- or 2r 

-12345 is -1- or 1- 

Because the decimal part of the logarithm is always positive, it 
is better to place the negative sign of the index above, instead of 
before it; thus, 3- instead of -3. For the log. of -00012345 is 
better expressed by 4-0914911, than by —4-0914911, because only 
the index is negative — {. 6., 4 is negative and -0914911 is positive, 
and may stand thus, —4- + -0914911. 

Sometimes, instead of employing negative indices, their comple- 
ments to 10 are used : 

for £-0914911 is substituted 6-0914911 

— 3^-0914911 7-0914911 

— 2-0914911 8-0914911 

&c. &c. 

When this is done, it is necessary to allow, at some subsequent 
stage, for the tens by which the indices have thus been increased. 

It is so easy to take logarithms and their corresponding numbers 
out of tables of logarithms, that we need not dwell on the method 
of doing so, but proceed to their application. 

MULTIPLICATION BY LOGARITHMS. 

Take the logarithms of the factors from the table, and add them 
together ; then the natural number answering to the sum is the 
product required : observing, in the addition, that what is to be 
carried from the decimal parts of the logarithms is always positive, 
and must therefore be added to the positive indices ; the difterence be- 
tween this sum and the sum of the negative indices is the index of the 
logarithm of the product, to which prefix the sign of the greater. 

This method will be found more convenient to those who have 
only a slight knowledge of logarithms, than that of using the arith- 
metical complements of the negative indices. 

1. Multiply 37-153 by 4-086, by logarithms. 

Nos. Logs. 

37-153 1-5699939 

4-086 .0-6112984 

Prod. 151-8071 .2-1812923 

2. Multiply 112-246 by 13-958, by logarithms. 

Nos. Logs. 

112-246 2-0501709 

13-958 1-1448232 

Prod. 1566-729 3-1949941 



336 THE PRACTICAL MODEL CALCULATOR. 

3. Multiply 46-7512 by -3275, by logaritbins. 

Nos. Logs. 

46-7512 1-6697928 

•3275 1-5152113 

Prod. 15-31102 1-1850041 

Here the +1 that is to be carried from the decimals, cancels 
the — 1, and consequently there remains 1 in the upper line to be 
set down. 

4. Multiply -37816 by -04782, by logarithms. 

Nbs. Logs. 

•37816 T-5776756 

•04782 2-6796096 

Prod. 0^0180836 .2-2572852 

Here the +1 that is to be carried from the decimals, destroys 
the —1 in the upper line, as before, and there remains the —2 
to be set down. 

5. Multiply 3^768, 2-053, and -007693, together. 

Nos. Logs. 

3-768 0-5761109 

2-053 0^-3123889 

-007693 '3-8860957 

Prod. -0595108 .2-7745955 

Here the +1 that is to be carried from the decimals, when ad- 
ded to —3, makes —2 to be set down. 

6. Multiply 3-586, 2-1046, -8372, and -0294, together. 

Nbs. Logs. 

3-586 .....0-5546103 

2-1046 0-3231696 

•8372 .1-9228292 

-0294 ¥-4683473 

Prod. -1857618 1-2689564 

Here the -f2 that is to be carried, cancels the —2, and there 
remains the — 1 to be set down. 

DIVISION BY LOGARITHMS. 

From the logarithm of the dividend, subtract the logarithm of 
the divisor ; the natural number answering to the remainder will be 
the quotient required. 

Observing, that if the index of the logarithm to be subtracted is 
positive, it is to be counted as negative, and if negative, to be con- 
sidered as positive ; and if one has to be carried from the decimals, 
it is always negative : so that the index of the logarithm of the 
quotient is equal to the sum of the index of the dividend, the index 



THE APPLICATION OF LOGARITHMS. 337 

of the divisor with its sign changed, and —1 when 1 is to be 
carried from the decimal part of the logarithms. 

1. Divide 4768-2 by 36-954, bj logarithms. 

Nbs. Logs, 

4768-2 3-6783545 

36-954 1-5676615 

Quot. 129-032 2-1106930 

2. Divide 21-754 by 2-4678, by logarithms. 

Nbs. Logs. 

21-754 1-3375391 

2-4678 ... 0-3923100 

Quot. 8-81514 0-9452291 

3. Divide 4-6257 by -17608, by logarithms. 

Nos. Logs. ^ 

4-6257 0-6651775 ^ 

•17608 1-2457100 

Quot. 26-27045 1-4194675 

Here the —1 in the lower index, is changed into +1, which is 
Chen taken for the index of the result. 

4. Divide -27684 by 5-1576, by logarithms. 

Nbs. Logs. 

•27684 .1-4422288 

5-1576 , 0-7124477 

Quot. -0536761 2"-7297811 

Here the 1 that is to be carried from the decimals, is taken as 
—1, and then added to —1 in the upper index, which gives —2 
for the index of the result. 

5. Divide 6-9875 by -075789, by logarithms. 

Nos. Logs. 

6-9875 0-8443218 

•075789 2-8796062 

Quot. 92-1967 1-9647156 

Here the 1 that is to be carried from the decimals, is added to 
—2, which makes —1, and this put down, with its sign changed, 
is +1. 

6. Divide -19876 by -0012345, by logarithms. 

Nos. Logs. 

•19876 r2983290 

-0012345 : 3-0914911 

Quot. 161-0043 2-2068379 

Here —3 in the lower index, is changed into +3,. and this ad- 
ded to 1, the other index, gives +3 — 1, or 2. 
2D 22 



338 THE PRACTICAL MODEL CALCULATOR. 

PROPORTION; OR, THE RULE OF THREE, BY LOGARITHMS. 

From the sum of the logarithms of the numbers to be multipiied 
together, take the sum of the logarithms of the divisors : the re- 
mainder is the logarithm of the term sought. 

Or the same may be performed more conveniently, for any 
single proportion, thus : — Find the complement of the logarithm 
of the first term, or what it wants of 10, by beginning at the left 
hand and taking each of the figures from 9, except the last figure 
on the right, which must be taken from 10 ; then add this result 
and the logarithms of the other two figures together: the sum, 
abating 10 in the index, will be the logarithm of the fourth term. 

1. Find a fourth proportional to 37425; 14-768, and 135-279, 
by logarithms. 

Log. of 37-125 1-5696665 

, Complement ,...8-4303335 

Log. of 14-768 1-1693217 

Log. of 135-279 2-1312304 

Ans. 53-8128 '. 1-7308856 

2. Find a fourth proportional to -05764, -7186, and -34721, by 
logarithms. 

Log. of -05764 2-7607240 

Complement 11-2392760 

Log. of -7186 1-8564872 

Log. of -34721 1-5405922 

Ans. 4-32868 0-6363554 

3. Find a third proportional to 12-796 and 3-24718, by logarithms, 

Log. of 12-796 1-1070742 

Complement 8-8929258 

Log. of 3-24718 0-5115064 

Log. of 3-24718 0-5115064 

Ans. -8240216 ' T-9159386 

INVOLUTION; OR, THE RAISING OF POWERS, BY LOGARITHMS. 

Multiply the logarithm of the given number by the index of 
the proposed power ; then the natural number answering to the 
result will be the power required. Observing, if the index be nega- 
tive, the index of the product will be negative ; but as what is to 
be carried from the decimal part will be affirmative, therefore the 
difference is the index of the result. 

1. Find the square of 2-7568, by logarithms. 

Log. of 2-7568 0-4404053 

2 

Square 7-599947 0-8808106 



THE ^APPLICATION OF LOaARITHMS. 339 

2. Find the cube of 7-0851, bj logarithms. 

Log, of 7-0851 0-8503460 

Cube 355-6625 2-5510380 

Therefore 355*6625 is the answer. 

3. Find the fifth power of -87451, by logarithms. 

Log. of -87451 r-9417648 

5 

Fifth power -5114695 1^7088240 

Where 5 times the negative index 1, being --5, and +4 to 
carry, the index of the power is 1. 

4. Find the 365th power of 1-0045, by logarithms. 

Log. of 1-0045 0-0019499 

365 

97495 
116994 

58497 

Power 5-148888 Log. 0-7117135 

EVOLUTION; OR, THE EXTRACTION OF ROOTS, BY LOGARITHMS. 

Divide the logarithm of the given number by 2 for the square 
root, 3 for the cube root, &c., and the natural number answering 
to the result will be the root required. 

But if it be a compound root, or one that consists both of a root 
and a power, multiply the logarithm of the given number by the 
numerator of the index, and divide the product by the denomina- 
tor, for the logarithm of the root sought. 

Observing, in either case, when the index of the logarithm is 
negative, and cannot be divided without a remainder, to increase 
it by such a number as will render it exactly divisible ; and then 
carry the units borrowed, as so many tens, to the first figure of the 
decimal part, and divide the whole accordingly. 

1. Find the square root of 27*465, by logarithms. 

Log. of 27-465 2 )1-4387796 

RoQt 5-2407 -7193898 

2. Find the cube root of 35-6415, by logarithms. 

Log. of 35-6415 3 ) 1-5519560 

Root 3-29093 ..-5173186 

3. Find the fifth root of 7-0825, by logarithms. 

Log. of 7-0825 5 ) 0-8501866 

Root 1-479235 ...-1700373 



340 THE PRACTICAL MODEL CALCULATOR. 

4. Find the 365th root of 1*045, by logarithms. 

Log. of 1-045 365) t)-0191163 

Root 1-000121 .0-0000524 

5. Find the value of (-001234)*, by logarithms. 

Loff. of -001234 3"-0913152 

_2 

3) 6-1826304 

Ans. -00115047 .2-0608768 

Here the divisor 3 being contained exactly twice in the negative 
index —6, the index of the quotient, to be put down, will be —2. 

6. Find the value of (-024554)^, by logarithms. 

Log. of -024554 .2-3901223 

2)64703669 

Ans. -00384754 ."3-5851834 

Here, 2 not being contained exactly in —5, 1 is added to it, 
which gives —3 for the quotient ; and the 1 that is borrowed being 
carried to the next figure makes 11, which, divided by 2, gives 
-5851834 for the decimal part of the logarithm. 

:method of calculating the logarithm of any given number, 
and the number corresponding to any gi^ten logarithm. dis- 
covered by olr^er byrne, the author of the present work. 

The succeeding numbers possess a particular property, which is 
worth being remembered. 

log. 1-371288574238542 = 0-1371288574238542 
log. 10-00000000000000 = 1-000000000000000 
log. 237-5812087593221 = 2-375812087593221 
log. 3550-260181586591 = 3-5o0260181586591 
log. 46692-46832877758 = 4-669246832877758 
log. 576045-6934135527 = 5-760456934135527 
log. 6834720-776754357 = 6-834720776754357 
log. 78974890-31398144 = 7-897489031398144 
log. 895191599-8267852 = 8-951915998267839 
log. 9999999999-999999 = 9-999999999999999 
In these numbers, if the decimal points be changed, it is evident 
the logarithms corresponding can also be set down without any cal- 
culation whatever. 

Thus, the log. of 137-1288574238542 = 2-1371288574238542; 
the log. of 35-50260181586591 = 1-550260181586591; 
log. -002375812087593221 = 3-^812087593221; 
log. -0008951915998267852 = 4-951915998267852 ; 



THE APPLICATION OF LOaARITHMS. 



341 



I 



and so on in similar cases, since the change of the decimal 
point in a number can only affect the whole number of its loga- 
rithm. 

These numbers whose logarithms are made up of the same digits 
will be found extremely useful hereafter. We shall next give a 
simple method of multiplying any number by any power of 11, 101, 
1001, 10001, 100001, &c. 

This multiplication is performed by the aid of coefficients of a 
binomial raised to the proposed power. 

(x -{- ^y — X + 2/, the coefficients are 1, 1. 

{x + ^f = x^ -h 2x^ + ^% the coefficients are 1, 2, 1. 

(x + ^Y = x^ '+ Zx^y H- ^xy"^ + t/^, the coefficients are 1, 3, 3 1. 

The coefficients of (a; + yj^'are 1, 4, 6, 4, 1. 

— •- \x-\'yY^ 1,5,10,10,5,1. 

— — {x^yf— 1, 6, 15, 20, 15, 6, 1. 

— — (a; -f yy— 1, 7, 21, 35, 35, 21, 7, 1. 

— — \x^yf— 1, 8, 28, b% 70, b^, 28, 8, 1. 
_ — \x-\-yf— 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. 

Let it be required to multiply 54247 by (101)^ 

The number must be divided into periods of two figures when the 
multiplier is 101 ; into periods of three figures when the multiplier 
is 1001 ; into periods of four figures when the multiplier is 10001 ; 
and so on. 



d 


c 


h 


a 






54 


24 


70 


00 


00 


1 


3 


25 


48 


20 


00 


a 6 




8 


13 


70 


50 


h 15 




10 


84 


94 


c 20 




8 


14 


d 15 




3 


e 6 



(54247) X (101)^ = 57 58 42 83 61, true to 10 places of figures. 

This operation is readily understood, since the multipliers for the 
6th power are 1, 6, 15, 20, 15, 6, 1 ; we begin at a^ a period in ad- 
vance, and multiply by 6 ; then we commence at 6, two periods in 
advance, and multiply by 15 ; at c, three periods in advance, and 
multiply by 20 ; at d^ four periods in advance (counting from the 
right to the left), and multiply by 15 ; the period, e, should be 
multiplied by 6, but, as it is blank, we only set down the 3 carried 
from multiplying c?, or its first figure by 6. 

As it is extremely easy to operate with 1, 5, 10, 10, 5, 1, the 
multipliers for the 5th power, it may be more convenient first to 
multiply the given number by (101)*, and then by (101)^ ; because, 
to multiply any number by 5, we have only to affix a cipher (or 
suppose it affixed) and to take the half of the result. 

The above example, if worked in the manner just described, will 
stand as follows : 
2d2 



S42 



THE PRACTICAL MODEL CALCULATOR. 





d 


c 


h 


a 








54 


24 


70 


00 


00 


1 




2 


71 


23 


50 


00 


5..« 






5 


42 


47 


00 


...10..5 






5142 


47 


...10..C 






2 


71 


5..d 






1 


1 


(54247) X (101)^ 


= 57 014114219 
57 Ol|41 42 





57 58 42 83 61 = (54247)^ x (101)^ 

The truth of this is readily shown by common multiplication, but 
the process is cumbersome. However, for the sake of comparison, 
we shall in this instance multiply 54247 by (101) raised to the 6th 
power. 

101 

101 

101 
1010 



10201 = (101)^. 
101 



10201 
102010 



1030301 = (lom 
101 



1030301 
10303010 

104060301 = (lom 
101 

104060401 
1040604010 

10510100501 = (101)'. 
101 

10510100501 
105101005010 

1061520150601 = (101)^ 
54247 



7430641054207 
4246080602404 
2123040301202 
4246080602404 
5307600753005 



575842836019652447 the required product, 



THE APPLICATION OF LOaARITHMS. 



343 



which shows that the former process gives the result true to 10 
places of figures, of which we shall add another example. 

Multiply 34567812 by (1001)^ so that the result may be true to 
12 places of figures. 



h 
3456 
2 



a 

78120000 

7654 2496 

96790 

i 19 



1 

8..a 

...28..5 
...56..(j 



3459 5475 9305 the required product. 

The remaining multipliers, 70, bQ, 28, 8, 1, are not necessary in 
obtaining the first 12 figures of the product of 34567812 by 10001 
in the 8th power. 

As 28 and bQ are large multipliers, the work may stand thus 

h 



3456 

2 



a 

7812 

7654 

6 

2 



0000 
2496 
9136 
7654 
17 
2 



1 

...a.. 8 
...5..20 
...5.. 8 
...C..50 
...c. 6 



28 



Result, = 345954759305 the same as before. 

Perhaps this product might be obtained with greater ease by first 
multiplying 34567812 by (10001)^ and the product by (10001)^ ; 
the operation will stand thus : 

345678120000 1 

172839060 5 

34568 10 

3 10 

345850093631 = 34567812 x (10001)^ 

103755298 3 

10376 3 



345954759305 = twelve places of the product of 
34567812 by (10001)^ x (10001)^ = (34567812) x (10001)«. 

• Although these methods are extremely simple, yet cases will oc- 
cur, when one of them will have the preference. 

Our next object is to determine the logarithms I'l ; 1*01 ; 1*001 ; 
1-0001; 1-00001; &c. 

It is well known that 



log. (1 + 7l) = M (ll 
M being the modulus, = 



Jn2 + ^n^ - In"" + \n' — ^n^ + &c.) 
•432944819032618276511289, &c. 



It is evident that when n is ^, ^^0, ^^^ ^ohoi &c., the calcula- 



tion becomes very simple. 



344 THE PKACTICAL MODEL CALCULATOR. 

M = -4342944819032518 
I M = -2171472409516259 
I M = -1447648273010839 
1 M = -1085736204758130 
I M = -0868588963806504 
i M = -0723824136505420 
^ M = -0720420788433217 
i M = -0542868102379065 
I M = -0482549424336946 
^M = -0434294481903252 

&c. &c., are constants employed to determine 
tlie logarithms of 11, 101, 1001, 100001, &c. 

To compute the log. of 1-001. In this case n = yoVu- 

M 

-f jQQ^ = -0004342944819033 positive 

- gfinry = -0000002171472410 negative 

-0004340773346623 

+ poop = -0000000001447648 positive 
-0004340774794271 

iM 

- QOQoy = -0000000000001086 negative 

-0004340774793185 

m 

+ r^QQY = -0000000000000001 positive 

-0004340774793186 = the log. of 1-001 ; 
true to sixteen places. 

It is almost unnecessary to remark, that, instead of adding and 
subtracting alternately, as above, the positive and negative terms 
may be summed separately, which will render the operation niore 
concise. 



Positive Terms. 

-0004342944819033 

1447648 

1 

+ -0004342945266682 
- 000000217473496 



Negative Terms. 

•0000002171472410 
1086 

•0000002171473496 



-0004340774793186 = log. 1-001. 

In a similar manner the succeeding logarithms may be obtained 
to almost any degree of accuracy. 



THE APPLICATION OF LOGARITHMS. 



345 



Log. 1-1 
1-01 

1-001 
1-0001 
1-00001 
1-000001 
1-0000001 
1-00000001 
1-000000001 
1-0000000001 
1-00000000001 
1-000000000001 
1-0000000000001 
1-00000000000001 
&c. 



= -041392685158225 &c. which we call A 



-004321373782643 
•000434077479319 
-000043427276863 
-000004342923104 
•000000434294265 
•000000043429447 
-000000004342945 
•000000000434295 
-000000000043430 
-000000000004343 
-000000000000434 
-000000000000043 
•000000000000004 
&c. 



B 



D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

&c. 



Without further formality or paraphernalia, for it is presumed 
that such is not necessary, we shall commence operating, as the 
method can be acquired with ease, and put in a clearer point of 
view by proper examples. 

Required the logarithm of 542470, to seven places of decimals. 



54 
3 



24 
25 

8 



70 
48 
13 
10 



2 
71 

85 



5758 



4284 = 6B 

7275 
3 



•02592824 



Take 57601562 = 3D = -00013028. 
From5 7 6 045 6 9 



576) 



310 7 
2880 = 5E 



•00002171 



127 

115 = 2F = -00000087 



12 

12 = 2G 



= -00000009 

-02608119 Take 
5-76045693 From 



Hence we have log. 542470.= 5-73437574, which is correct 
to seven decimal places. 

6B is written to represent 6 times the log. of 1-01. 

The nearest number to 542470, whose log. is composed of the 
same digits as itself, being 576045*6934, &c., our object was to 
raise 542470- to 576045*69 by multiplying 542470* by some power 
or powers of 1*1, 1*01, 1*001, 1*0001, &c. 



346 



THE PRACTICAL MODEL CALCULATOR. 



It is here necessary to remark, that A is not employed, because 
the given number multiplied by 1*1, would exceed 576045*69 ; for 
a like reason C is omitted. 

Again, when half the figures coincide, the process may be per- 
formed (as above) by common division ; the part which coincides 
becoming the divisor ; thus, in finding 5 E, 576 is divided into 3007, 
it goes 5 times, the E showing that there are five figures in each 
period at this step. For A, there is but one figure in each period ; 
for B, there are two figures ; for C, there are three figures in each 
period, and so on. 

Let it be required to calculate the logarithm of 2785*9, true to 
seven places of decimals. 

It will be found more convenient, in this instance, to bring the 
given number to 3550*26018, the log. of which is 3*55026908. 



OjO 



33 
1 



70 

68 

3 



9390 = 2A = *08278537 



54 

37 

3 



70 
9 
37 

2 



354 



289 
708 



08 = 5B = -02160687 

58 

35 



Take 3549 
From 3 5 5 



9801 = 20 = *00086815 
2602 



355) 



21801 = 7E = *00003040 

2485 



16 = 8F = *00000347 

84 



2 = 9 G- = *00000039 
2 . 

Take *10529465 
From 3*55026018 



log. 2785*9 = 3*44496553 

At the Observatory at Paris, g = 9*80896 metres, the second 
being the unit of time, what is the logarithm of 9*80896 ? 
In this example, we shall bring 9*80896 to 9*99999, &c. 



THE APPLICATION OF LOGARITHMS. 



347 



9 810 8 

98 



9 6,0 

08 9 6 



00 
00 



990704 

8|916 

35 



960 
344 

83 



= 1 B = -0043213738 



99965705 

"2:9 9 8 9 

3 



32 = 90 = 

72 
00 

3D = 



•0039066973 



•0001302818 



99995169804 

39 99 8 3 

6 

Take 9999969793 = 4E= -0000173717 
From 10000000000 



30207 

From which we have 3 F 

2H 



-0000013029 
•0000000087 



7 J = -0000000003 

Take -0083770365 
From 1-0000000000 

Log. 9-80896 = -9916229635 
As before observed, 9 C might have been obtained in the follow- 



ing manner 



8 9 017 04 

4|9 5 3 

9 



960 

524 

907 

9 



= 1 B, as above. 



5 times 9 9 516 6 8 

3|982 
5 



401 
673 

973 
4 



4times9996570532 = 9 C. 

A French metre is equal to 3-2808992 English feet, required 
the log. of 3-2808992. 



e d 


c 


b 


a 




32 


80 


89 


92 


00., .once 


2 


29 


66 


29 


44... 7 times from a 




6 


88 


98 


88. ..21 — b 




11 


48 


31...35 — c 




11 


48. ..35 — d 




7. ..21 — e 



3517 56 8018 = B 7. 



i 



348 THE PRACTICAL MODEL CALCULATOR. 

The manner in which B 7 is obtained is worthy of remark : the 
multipliers being 1, 7, 21, 35, 35, 21, 7, 1, when 7 times the first 
line (commencing with the period marked a) is obtained, 21 times 
the same line (commencing with the period marked h) is determined 
by multiplying the 2d line by 3. If the 2d line be again multiplied 
by 5, we have the 4th line of the multiplier 35; but to multiply 
by 5, we have only to take the half the product produced by mul- 
tiplying by 7, advancing the result one figure to the right. Hence, 
to find the result for 35 is almost as easy as to find the result 
for 5. 

But the object in this case being to bring the proposed number 
to 35502601815, the process must be continued. 

h 



c 
1 351 
9 3 

36 



a 

7561801 



8 = B 7, as above. 
2 
2 
84 29 6 



165 811 2 
12 663 2 



354 935 305 8 = C 9 

The 2d (or 9) line is produced by beginning at «, but the multi- 
plication may be performed by subtracting 3517568 from 35175680 ; 
the 36 line is produced by beginning at 6, observing to carry from 
the preceding figure, making the usual allowance when the number 
is followed by 5, 6, 7, 8, or 9. The 36 line may be produced by 
multiplying the 9 line by 4, beginning one period more to the left. 
To multiply by 84 is not apparently so convenient, for 84 X 352 = 
29|568 ; and as only one figure of the period 568 is required, when 
the proper allowance is made, the result becomes 29|6. 

But, since 84 is equal to 36 X 2J, we have only to multiply the 
36 line by 2, and add J of it ; with such management, tjie work 
will stand thus : — 



351 756 

31165 

12 



801 
811 
663 



8 = B 7, as before 
2 = 9 times 
2 = 36 times 



243 = 72 times \ _ o. ,. ,^ 
4'2 = 12 times \ = ^^ *^^^^ 



354 935 305 8 = C 9 
This amounts to very little more than adding the above numbers 
together. 

Many other contractions will suggest themselves, when the mul- 
pliers are large : thus, to multiply any number 57837 by 9, as 
alluded to above, is easily efi'ected, by the following well-known 
process : — Subtract the first figure to the right from 10, the second 
from the first, the third from the second, and so on. 

r 578370... ten times 
Thus, 57837 x 9 =^ 57837 . ..once 

(^520533... nine times 



THE APPLICATION OF LOGARITHMS. 



349 



Such simple observations are to be found in every book on men- 
tal arithmetic, and therefore require but little attention here. 
The whole work of the previous example will stand thus : — 



32 

2 



8 018 9 9200 



2 96 6 

6|8 8 

11 



29 
98 
48 
11 



44 

88 
31 
48 + 7 



B7 = 351 

3 



756 

165 

12 



801 
811 

663 
29 



8 = -0302496165 

2 
2 
6 



7B 



C9 = 354 913 530 

70 9 8 



5 8 = -0039066973 = 90 

71 

35 



D2 = 3550 0j6 2964 = -0000868546 = 2 D 
1!7 7 5 3 
4 



Take E5 = 3550240471 = -0000217146 = 5 E 
From 3550260182 



3550) 
F5 



G5 
H5 



9711 

7750 



0000021715 = 5 F 
5G 



1|9 61 

IjT 7 5 = -0000002172 



7 8 = -0000000217 = 5 H 



12 



J a 



7 = -0000000009 :^ I 2 



•0000000001 = J 3 



Take -0342672944 
From 3-5502601816 

Log. 3280-8992 = 3-5159928972 
.-. log. 3-2808992 = 0-5159928972. 

The constant sidereal year consists of 365-25636516 days ; what 
is the log. of this number ? 

In this case it is better to bring the constant 35502601816 to 
36525636516, instead of bringing the given number to the con- 
stant, as in the former examples. 
2E 



350 



THE PRACTICAL MODEL CALCULATOR. 



3 5|5 0|2 6 

|7 l|0 
35 



B2 = 362 

2 



162 

897 

10 



1'8 1 



52 
50 



03 

26 



041 
296 
140 

20 



12 
33 
54 

28 



= -0086427476 = 2 B 



C8 = 3650 
1 



6949827 = -0034726298 = 8 C 
8253475 
6 51 



Take D5 = 36525206953 = -0002171364 = 5 D 
From 36525636516 



36525-2) 
El = 

Fl = 

G7 = 



429563 
365252 = 



-0000043429 = 1 E 

64311 

3 6 5 2 5 = -0000004343 = 1 F 



H6 = 
10 
J7 = 



217 7 8 6 
2 I5 5 6 8 

2 

2 



-0000003040 = 7 0- 



218 

191 = -0000000261 



6H 



2,7 

2i5 = -0000000003 = 7 J 

-0123376214 
Add 3-5502601816 
Hence, log. 3652-5636516 = 3-5625978030 
.-. log. 365-25636516 = 2-562597803. 
M. Regnault determined with the greatest care the density of 
mercury to be 13-59593 at the temperature 0°, centigrade. It is 
required to calculate the log. of 13-59593, to eight places of decimals. 
In this case it is better to bring the given number to the constant 



1371288574. 



1 3 5;9 5 9 

110 8 7 
3 



300 
674 
807 



C8 = 137 0,5 078 

16 8 52 
1 



8 = -003472630 = 80 

5 

4 



Subtract D5 = 137119328 = -000217136 = 5 D 
From 137128857 

9,5 2 9 = -000026058 = E 6 
E6 = 812 2 7 
1310 2 
F 9 = 1 2I34 = -000003909 = F 9 

6|8 
H5= 6l9 = -000000022 = H 5 
•003719755 



APPLICATION OF LOGARITHMS. 351 

Take -003719755 
From -137128857 

log. 1-359593 = -133409102 
.-. log. 13-59593 = 1-133409102. 

TO DETERMINE THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 

This problem has been very much neglected — so much so, that 
none of our elementary books ever allude to a method of comput- 
ing the number answering to a given logarithm. When an opera- 
tion is performed by the use of logarithms, it is very seldom that 
the resulting logarithm can be found in the table ; we have, there- 
fore, to find the nearest less logarithm, and the next greater, and 
correct them by proportion, so that there may be found an inter- 
mediate number that will agree with the given logarithm, or nearly 
so. But although the i^'^^oportional j^cirts of the difference abridge 
this process, we can only find a number appertaining to any loga- 
rithm to seven places of figures when using our best modern tables. 
As, however, the tabular logarithms extend only to a degree of 
approximation, fixed generally at seven decimal places, all of which, 
except those answering to the number 10 and its powers, err, either 
in excess or defect, the maximum limit of which is J in the last 
decimal, and since both errors may conspire, the 7th figure cannot 
be depended on as strictly true, unless the proposed logarithm falls 
between the limits of log. 10000 and log. 22200. 

Indubitably we are now speaking of extreme cases, but since it 
is not an unfrequent occurrence that some calculations require the 
most rigid accuracy, and many resulting logarithms may be ex- 
tended beyond the limits of the table, this subject ought to have 
a place in a work like the present. It is not part of the present 
design to enter into a strict or formal demonstration of the follow- 
ing mode of finding the number corresponding to a given logarithm, 
as the operation will be fully explained by suitable examples. 

What number corresponds to the logarithm 3*4449555 ? 

The next less constant log. to the one proposed is 2-37581209, 
or rather, 3*37581209, when the characteristic or index is increased 
by a unit. Secondly. 



First from 3*44496555 
take 3-37581209 



37581209 constant 
23758121 = Al 



-06915346 2 6 1 3;3 9 3 3 



'04139269 == lA 115 6 8 

•02776077 ^'^ ^ 

•02 592824 = 6 B ^ 

7^183253 

173631 = 4 C 



36 

00 

22 

3 



277 
1 



416 
109 



...9622 1 



965 = B6 

668 
664 



8685 = 2 D 1 

937 27 8 5|2 8 2 918 = C 4 



352 



THE PKACTICAL MODEL CALCULATOR. 



937 

869 

.68 
43 



2E 



= 1F 



27852 82 918 = 04 
155 7 06 
|3 
T=D2 



27858 



25 

22 = 5G 

.3 

3 



410 

5|5 
|2 



1^3 



2 = E2 
9 = F1 

9 = G5 



= 7H 



1'9 = H7 



278590016 
.♦. 2785-90016 is the number sought. 
What number corresponds to the logarithm 5-73437574 ? 
"When the index of this log. is reduced by a unit, the nearest 
next less constant is 4-66924683. 
From 4-73437574 
Take 4-66924683 
•6512891 

4139269 . 1 A 

•2373622" 

2160687 5B 

• • 212035 
173631... 



.4 



39304 

39085. 



.9D 



219 There is neither the equal of 
217 5 F this number, nor a 



2 G less, obtainable from 

2 4 H E, .-. EO, or E, is 

- omitted. 

Then, 416 6 9246 8 3 

46692468 Al 



5 113 611 7|1 51 

215 6 8 0:8 5 8 

5113,617 

513 6 

26 



5 3 918 1 6 

2|l5 9 

3 



7 8 8. 
267 
239 

2 



.B5 



5419i792 9 

4l8 7 7 8 
19 

;7 



5424672 

27 



.0 4 



.D9 
.F5 
.H4 



542470006 
542470-006 is the number whose logarithm is 5-73437574. 



THE APPLICATION OF LOGARITHMS. 



353 



Had the given logarithm represented a decimal with a positive 
index, the required number would be 0'0000o4247, &c. ; or if 
written with a negative index, as 5-73437574, the result would be 
the same, for the characteristic 5, shows how many places the first 
significant figure is below unity. 

Required the number corresponding to log. 2*3727451. 
The constant 100000000 is the one to be employed in this case. 
1*3727451 the given log. minus 1 in the index. 
1*0000000 



*3727451 
3725342. 

. . . 2109 
1737 . 

. ... 372 
347 . 

25 

22. 



,9 A 



.4D 



.8E 



.5F 



3 

i' 
llOIOiO 

|9l0 

3'6 

'8 4 

12 

1 



,7G 



Constant. 













6 



2357 



9485 

9432 

1 



2 358 8j9 

18 

1 



A9 



D4 

E8 
F5 

G7 



23590949 
.*. 235*90949 is the required number, and the seconds in the di- 
urnal apparent motion of the stars. 

235-90949'' = 3' 55-90949". 

Let it be required to find the hyperholio logarithm of any 

number, as 3*1415926536. The common log. of this number is 

•49714987269 (33), and the common log. of this log. isT-6964873. 

The modulus of the common system of logarithms is -4342944819, 

&c. 

.-. 1 : 4342944819 : : hyperbolic log. N : common log. N. 
2b2 23 



354 



THE PRACTICAL MODEL CALCULATOR. 



To distinguish the hyperbolic logarithm of the number N from 
its common logarithm, it is necessary to write the hyp. log. Log. N, 
and the common logarithm log. N. * 

Hence, 4342944819 x Log. N = log. N ; 
or log. (-4342944819) + log. (log. N) = log. (lo_g. N). 
.-. log. (Log. N) = log. (log. N) - 1-6377843 ; for 1-6377843 = 
log. -4342944819. 

Now, to work the above example, from 1*6964873 

take r6377843 

•0587030, the number 
corresponding to this com. log. will be the hyp. log, of 3-1415927. 
•0587030 must be reduced to -0000000 which is known to be the 
log. of 1. 



•0587030 
0413927 

. 173103 

172855 

.... 248 

217 

31 

30 
1 



1 A = 11 



1 A 

4B 
5E 

7F 

2G 



00 
44 



010 0|0 

6 60 

14 4 
1 



144 616 

5 



4|4 
7,2 

8;o 

2 



= G2 



114472988 

.-. 1-14472988 is the hyperbolic log. of 3-1415927, true to the 
last figure ; for the hyp. log. 3-1415926535898 = 1-1447298858494. 

The reason of this operation is very clear, because 
1 X 1-1 X (1-01)^ X (1-00001)^ X (1-000001)^ x (1-0000001^ = 
1-14472988. 

This example answers the purpose of illustration, but the hyp. 
log. of 3-1415927 can be more readily found by dividing its com. 
log. -49714987269 by the constant -4342944819, which is termed 
the modulus of the common system of logarithms. 

Suppose it is known that 1-3426139 is the log. of the decimal 
which a French litre is of an English gallon. Required the decimal. 

The index, 1, may be changed to any other characteristic, so as 
to suit any of the constants, as the alteration is easily allowed for 
when the work is completed. In this instance, it is best to put 
+1 instead of 1. 



From 
Take 



1-3426139 

1-0000000 

-3426139 

3311415 

•0114724 

. .86427 

28297 

26045 

2252 



= 8 A 



= 2B 



= 6C 



IjOO 

8 

2 8 

5 



00 



0,0 

o'o 

60 

70 

5 



Constant 








6 00 
2800 
80 

i 

2 1|4 3|5 8|8 8|1 = AS 



00 
00 

o:o 

00 


80 



THE APPLICATION OF LOGARITHMS. 



355 



2252 




2114358^ 


B8 


1 = A8 


2171 = 


5D 


4 28 71718 


81 


IE 


21^ 


i3|6 


43 = 


218667495 = B 2 


38 




1312005 


35 = 
3 


8F 
7 G 


3280 
4 


3 = 


2 1 9 9:8 2 7 8 


4 = C6 






I'O 9 9 9 


1 






' 2 


2 




2 2 9-2,719 7 = D 5 






|2^ 


2 1 = I) 1 






1761 = F 8 






754 = G7 



220096913 
.-. The French litre = -2200969 English gallons. 
In measuring heights by the barometer, it is necessary to know 
the ratio of the density of the mercury to that of the air. 

At Paris, a litre of air at 0° centigrade, under a pressure of 760 
millimetres, weighs 1-293187 grammes. At the level of the sea, 
in latitude 45°, it weighs 1-292697 grammes. A litre of water, 
at its maximum density, weighs 1000 grammes, and a litre of mer- 
cury, at the temperature of 0° cent., weighs 13595-93 grammes : 
13595-93 
.-. -1.0Q2697 ~ ^'^^^^ ^* 
Now, log. 13595-93 = 4-133409102 (29) 
and log. 1-292697 = 0- 111496744 (30) 

4^021912358 = the log. of the ratio at 45°. 

To find the number corresponding to this log. , it is necessary to reject 

the index for the present, and reduce the decimal part to zero. By 

this means the necessity of using any of the constants is superseded. 

-021912358 

•021606869 



305489 
303991 



= 5B 



= 7D 



io|oooooo;oo 

00^ ^ 



'5 

1 



00 

1 



1498 

1303 = 3 F 



oo;oo 
0:0 

00 
50 



1 5 I'O 1 05 = B 5 



195 

174 = 4 a 



17357 
2 



105174 



21 

17 = 4H 

4 

4 = 91 

13595-93 
.-. by logarithms, -^.292^97 

verified by common division. 



59,8 
3l|6 

4|2 
4 
1 



D7 
F 3 
G4 
H4 
19 



105174961 



= 10517-49, &c., which is easily 



356 



THE PRACTICAL MODEL CALCULATOR. 



W = 



M. Regnault found that, at Paris, the litre of atmospheric air 
weighs 1*293187 grammes ; the litre of nitrogen 1*256167 grammes; 
a litre of oxygen, 1*429802 grammes; of hydrogen, 0*089578 
grammes ; and of carbonic acid, 1*977414 grammes. But, strictly 
considered, these numbers are only correct for the locality in which 
the experiments were made ; that is for the latitude of 48° 50' 14'' 
and a height about 60 metres above the level of the sea ; M. Keg- 
nault finds the weight of the litre of air under the parallel of 45° 
latitude, and at the same distance from the centre of the earth as 
that which the experiments were tried, to be 12*926697. 

Assuming this as the standard, he deduces for any other latitude, 
any other distance from the centre of the earth, the formula, 

1*292697 (1*00001885) (1 - 0*002837) cos. 2 x 

. 2A 

Here, w is the weight of the litre of air, R the mean radius of 
the earth = 6366198 metres, h the height of the place of observa- 
tion above the mean radius, and ^ the latitude of the place. 

At Philadelphia, lat. 39° 56' 51*5", suppose the radius of the 
earth to be 6367653 metres, the weight of the litre of air will be 
1*2914892 grammes. The ratio of the density of mercury to that 
of air at the level of the sea at Philadelphia is 10527*735 to 1 ; 
required the number of degrees in an arc whpse length is equal to 
that of the radius. 

360 
As 3*1415926535898 : 1 : : -y- : the required degrees. 

Log. 360 = 2*556302500767 
log. 3*14159265359 = 0*497149872694 

2*059452623073 
log. 2 = 0*301029995664 

1*758122632409 
number required. 

When the index of this log. is changed into 4, the nearest next 
less constant is 4*669246832878. 



= the log. of the 



From 4*758122632409 

Take 4*669246832878 

*088875799531 

2 A = *82785370316 



4|6|6j9 

9|3|3 
46 



813 

36 
46 



8 = Constant 

6 

9 



IB = 



40 = 



7 E = 



. 6090429215 
4321373783 

.1769055432 
1736309917 

...32745515 
30400462 



5 6|4 9,7 8 8 6:6 7 
5 649 7 88 6 



7 8,3 
6 78 



A2 



5 7 016 2 8 

2|2 8 2 

3 



655 
514 

423 

2 



446 
621 
771 

282 



1 = B1 

8 
9 
5 



2345053 57291|45961|229 = C4 



THE APPLICATION OP LOGARITHMS. 357 

2345053 5 7 2 9 114 5 9 6 112 2 9 = C 4 

'5r= 2171471 41010 401217 

173582 12j0 31 

3G= 130288 5729547013477 = E 7 

:43294 21864773 

9 H = 39087 ^ 



7^^ 5729575 

91 = 3909 1 



616 

71 



298 

6J = 261 



511 
5 
4 
37 4 



8K= 35 



2,69 = F5 
8173 = 03 
62 = H9 
66 = 19 
38 = J 6 
58 = K8 
29 = L5 



2 5729577951295 = the num- 

5 L = 2 ber required. 

But the original index is 1; .-. 57*29577951295° are the num- 
ber of degrees in an arc the length of which is equal to that of the 
radius. 

The above result may be easily verified by common division, a 
method, no doubt, which would be preferred by many, for loga- 
rithms are seldom used when the ordinary rules of arithmetic can 
be applied with any reasonable facility. However, this example, 
like many others, is introduced to show with what ease and correct- 
ness the number corresponding to a given log. can be obtained. 
The extent, also, by far exceeds that obtainable by any tables 
extant. 

Other computations give, 

r° = 57-2957795130° = 57° 17' 44" -80624 
the degrees in an arc = radius. 

/ = 3437-7467707849' = 3437' 44" -80624 
the minutes in an arc = radius. 

r" = 206264-8062470963 
the number of seconds in an arc = radius. 

The relative mean motion of the moon from the sun in a Julian or 
fictitious year, of 365|- days, is 12 cir. 4 signs, 12° 40' 15-977315' = 
16029615-977315". 

.-. 16029615-977315" : 1 circumference (= 129600") 
: : 365-25 days 
: 29-5305889216 days = the mean synodic month. 

This proportion may, for the sake of example, be found by loga- 
rithms. 

Log. 365-25 2-56259022460634 

log. 1296000 6-11260500153457 

8*67519522614091 
log. 16029615-977315 = 7-20492311805406 

1-47027210808685 



858 



THE PEACTICAL MODEL CALCULATOR. 



If the index of this log. be made 2 instead of 1, the nearest next 
less constant will be 2-375812087593221. 



From 2-47027210808685 
Take 2-37581208759322 

•09446002049363 
08278537031645 



2 A = 
2B = 
60 = 
9D = 
8E = 
4F = 
2G = 
5H = 
71 = 
IJ = 
5K = 

8L = 

2N = 



1167465017718 

864274756529 

. 303190261189 
260446487591 

..42743773598 

39084549177 

. . . 3659224421 
3474338483 

.... 184885938 
173717706 



11168232 

8685889 

.2482343 
2171473 

. . 310870 
304006 

6863 

4343 

.... 2520 
2172 

348 

34T 



3i7,5|8 

4:7j51 
'2!37 



120 

6:2:4 

5;8i 



7|5,9 3|2|2 Const. 



1171511 8i6 4 
,2i0;8,7 5!9;3 



28174 

57 



73 
49 

28 



2 6i2 5 9 8j7 7 
46 5 2'51|9 7 



74:7 32 6 



260 



293 
1 



251 

759 
4 



475 

508 

398 

5 



177 

851 

772 

865 

4 



015 

062 
128 
029 
399 

2 



2A 



2B 



29 5011538 

2|6 5 51 

10 



8669 

3849 

6205 

24 



635 

803 

540 

781 

4 



29528 
2 



1008 714 9763 

3622480700 

8126 7 87 

17 



29530 4|6 32057 
l|l81218 

1 



267 

528 

772 



C6 



D9 



E8 



2953058 



132775 617 = F4 
5906116i3 
8 



29530587 
1 



3;3 

76 

06 

2 

1 



8i7 3 
52 9 
7!l4 
9|5 3 
4:7 6 
2'3 6 



3 = G2 

4 = H5 
1=17 

1 = J1 

5 = K5 

2 = L8 
6=N2 



295305889217832 
.-. 29-5305889218 is the number required. 
To perform, by logarithms, the ordinary operations of multipli- 
cation, division, proportion, or even the, extraction of the square 
root, except in the way of illustration, is not the design of these 
pages ; for such an application of logarithms, in a particular man- 
ner only, diminish the labour of the operator. It is not necessary, 
however, to examine minutely here the instances in which common 
arithmetic is preferable to artificial numbers ; besides, much wiK 
depend on the skill and facility of the operator. 



359 



TRIGONOMETRY. 

ANGULAR MAGNITUDES. — TRIGONOMETRY. — HEIGHT AND DISTANCES. — 

SPHERICAL TRIGONOMETRY. THE APPLICATION OP LOGARITHMS TO 

ANGULAR MAGNITUDES. 

Plane trigonometry treats of the relations and calculations of 
the sides and angles of plane triangles. 

The circumference of every circle is supposed to be divided into 
360 equal parts, called degrees ; also each degree into 60 minutes, 
each minute into 60 seconds, and so on. 

Hence a semicircle contains 180 degrees, and a quadrant 90 de- 
grees. 

The measure of any anele is an arc of any circle contained be- 
tween the two lines which form that angle, the angular point being 
the centre ; and it is estimated by the number of degrees contained 
in that arc. 

Hence, a right angle being measured by a quadrant, or quarter 
of the circle, is an angle of 90 degrees ; and the sum of the three 
angles of every triangle, or two right angles, is equal to 180 de- 
grees. Therefore, in a right-angled triangle, taking one of the 
acute angles from 90 degrees, leaves the other acute angle ; and 
the sum of two angles, in any triangle, taken from 180 degrees, 
leaves the third angle ; or one angle being taken from 180 degrees, 
leaves the sum of the other two angles. 

Degrees are marked at the top of the figure with a small °, mi- 
nutes with \ seconds with ", and so on. Thus, 57° 30' 12" de- 
note 57 degrees 30 minutes and 12 seconds. 

The complement of an arc, is what it wants of 
a quadrant or 90°. Thus, if AD be a quadrant, 
then BD is the complement of the arc AB ; and, 
reciprocally, AB is the complement of BD. So 
that, if AB be an arc of 50°, then its complement 
BD will be 40°. 

The supplement of an arc, is what it wants of 
a semicircle, or 180°. Thus, if ADE be a semicircle, then BDE 
is the supplement of the arc AB ; and, reciprocally, AB is the sup- 
plement of the arc BDE. So that, if AB be an arc of 50°, then 
its supplement BDE will be 130°. 

The sine, or right sine, of an arc, is the line drawn from one 
extremity of the arc, perpendicular to the diameter passing through 
the other extremity. Thus, BF is the sine of the arc AB, or of 
the arc BDE. 

Hence the sine (BF) is half the chord (BG) of the double arc 
(BAG). 

The versed sine of an arc, is the part of the diameter intercepted 
between the arc and its sine. So, AF is the versed sine of the arc 
AB, and EF the versed sine of the arc EDB. 




360 



THE PRACTICAL MODEL CALCULATOR. 



The tangent of an arc is a line touching the circle in one ex- 
tremity of that arc, continued from thence to meet a line drawn 
from the centre through the other extremity : which last line is 
called the secant of the same arc. Thus, AH is the tangent, and 
CH the secant, of the arc AB. Also, EI is the tangent, and CI 
the secant, of the supplemental arc BDE. And this latter tangent 
and secant are equal to the former, but are accounted negative, as 
being drawn in an opposite or contrary direction to the former. 

The cosine, cotangent, and cosecant, of an arc, are the sine, 
tangent, and secant of the complement of that arc, the co being 
only a contraction of the word complement. Thus, the ares AB, 
BD being the complements of each other, the sine, tangent or se- 
cant of the one of these, is the cosine, cotangent or cosecant of the 
other. So, BF, the sine of AB, is the cosine of BD ; and BK, 
the sine of BD, is the cosine of AB : in like manner, AH, the 
tangent of AB, is the cotangent of BD ; and DL, the tangent of 
DB, is the cotangent of AB : also, CH, the secant of AB, is the 
cosecant of BD ; and CL, the secant of BD, is the cosecant of AB. 

Hence several remarkable properties easily follow from these 
definitions ; as, 

That an arc and its supplement have the same sine, tangent, and 
secant ; but the two latter, the tangent and secant, are accounted 
negative when the arc is greater than a quadrant or 90 degrees. 

When the arc is 0, or nothing, the sine and tangent are nothing, 
but the secant is then the radius CA. But when the arc is a 
quadrant AD, then the sine is the greatest it can be, being the ra- 
dius CD of the circle ; and both the tangent and secant are infinite. 

Of any arc AB, the versed sine AF, 
and cosine BK, or CF, together make 
up the radius CA of the circle. The 
radius CA, tangent AH, and secant 
CH, form a right-angled triangle CxlH. 
So also do the radius, sine, and cosine, 
form another right-angled triangle 
CBF or CBK. As also the radius, 
cotangent, and cosecant, another right- 
angled triangle CDL. And all these 
right-angled triangles are similar to 
each other. 

The sine, tangent, or secant of an 
angle, is the sine, tangent, or secant 
of the arc by which the angle is mea- 
sured, or of the degrees, &c. in the same 
arc or angle. 

The method of constructing the scales 
of chords, sines, tangents, and secants, 
usually engraven on instruments, for 
practice, is exhibited in the annexed 
figure. 




TRIGONOMETRY. 361 

A trigonometrical canon, is a table exhibiting the length of the 
sine, tangent, and secant, to every degree and minute of the quad- 
rant, with respect to the radius, which is expressed by unity, or 1, 
and conceived to be divided into 10000000 or more decimal parts. 
And further, the logarithms of these sines, tangents, and secants 
are also ranged in the tables ; which are most commonly used, as 
they perform the calculations by only addition and subtraction, 
instead of the multiplication and division by the natural sines, &c., 
according to the nature of logarithms. 

Upon this table depends the numeral solution of the several 
cases in trigonometry. It will therefore be proper to begin with 
the mode of constructing it, which may be done in the following 
manner : — 

To find the sine and cosine of a given arc. 

This problem is resolved after various ways. One of these is as 
follows, viz. by means of the ratio between the diameter and cir- 
cumference of a circle, together with the known series for the sine 
and cosine, hereafter demonstrated. Thus, the semi-circumference 
of the circle, whose radius is 1, being 3-141592653589793, &c., 
the proportion will therefore be, 

As the number of degrees or minutes in the semicircle. 
Is to the degrees or minutes in the proposed arc, 
So is 3-14159265, &c., to the length of the said arc. 
This length of the arc being denoted by the letter a; also its 
sine and cosine by s and c ; then will these two be expressed by the 
two following series, viz. : — 

a^ a^ a' 

^ "" ^ ■" 2:3"^ 2X4:5 ■" 2.3.4.5.6.7 "^ ^^* 
a? aJ' cC 

= ^ ■" "6 + 120 ~ 5040 + ^^• 
a"- a^ a^ 

^ = ^ ~ ^ "^ 2X4 ~ 2.-3.4.5.6 + ^^• 

a"- a^ a^ 

= ^-^ + 24-720 + ^^- 

«• 

If it be required to find the sine and cosine of one minute. 
Then, the number of minutes in 180° being 10800, it will be first, 
as 10800 : 1 : : 3-14159265, &c. : -000290888208665 = the length 
of an arc of one minute. Therefore, in this case, 

a = -0002908882 
and la^ = -000000000004, &c. 
the difference is s = -0002908882 the sine of 1 minute. 
Also, from 1- 

take \a^ = 0-0000000423079, &c. 
leaves c = -9999999577 the cosine of 1 minute. 
2 F 



362 THE PRACTICAL MODEL CALCULATOR. 

For the sine and cosine of 5 degrees. 

Here, as 180° ; 5° : : 3-14159265, &c., : -08726646 = a the 
length of 5 degrees. 

Hence, a = -08726646 

_ 1^3 == _ -00011076 
+ rk«' = -00000004 
these collected give § — -08715574 the sine of 5°. 

And, for the cosine, 1 = 1* 

- ja^ = _ .00380771 
+ ^a^ = -00000241 

these collected, give c = -99619470 the consine of 5°. 

After the same manner, the sine and cosine of any other arc 
may be computed. But the greater the arc is, the slower the series 
will converge, in which case a greater number of terms must be 
taken to bring out the conclusion to the same degree of exactness. 

Or, having found the sine, the cosine will be found from it, by 
the property of the right-angled triangle CBF, viz. the cosine 
CF = v/CB^ - BF^ or c = ^/l - sK 

There are also other methods of constructing the canon of sines 
and cosines, which, for brevity's sake, are here omitted. 

^ To compute the tangents and secants. 

The sines and cosines being known, or found, by the foregoing 
problem ; the tangents and secants will be easily found, from the 
principle of similar triangles, in the following manner : — 

In the first figure, where, of the arc AB, BF is the sine, CF or 
BK the cosine, AH the tangent, CH the secant, DL the cotangent, 
and CL the cosecant, the radius being CA, or CB, or CD; the 
three similar triangles CFB, CAH, CDL, give the following pro- 
portions : 

1. CF : FB : ; CA : AH ; whence the tangent is known, being 
a fourth proportional to the cosine, sine, and radius. 

2. CF : CB : : CA : CH; whence the secant is known, being a 
third proportional to the cosine and radius. 

3. BF : FC : : CD : DL ; whence the cotangent is known, being 
a fourth proportional to the sine, cosine, and radius. 

4. BF : BC : : CD : CL; whence the cosecant is known, being 
a third proportional to the sine and radius. 

Having given an idea of the calculations of sines, tangents, and 
secants, we may now proceed to resolve the several cases of trigo- 
nometry ; previous to which, however, it may be proper to add a 
few preparatory notes and observations, as below. 

There are usually three methods of resolving triangles, or the 
cases of trigonometry — namely, geometrical construction, arith- 
metical computation, and instrumental operation. 

In the first method. — The triangle is constructed by making the 
parts of the given magnitudes, namely, the sides from a scale of 



TRIGONOMETRY. 363 

equal parts, and the angles from a scale of chords, or by some 
other instrument. Then, measuring the unknown parts by the 
same scales or instruments, the solution will be obtained near 
the truth. 

In the second method. — Having stated the terms of the propor- 
tion according to the proper rule or theorem, resolve it like any 
other proportion, in which a fourth term is to be found from three 
given terms, by multiplying the second and third together, and 
dividing the product by the first, in working with the natural num- 
bers ; or, in working with the logarithms, add the logs, of the 
second and third terms together, and from the sum take the log. 
of the first term ; then the natural number answering to the re- 
mainder is the fourth term sought. 

In the third method. — Or instrumentally, as suppose by the log. 
lines on one side of the common two-foot scales ; extend the com- 
passes from the first term to the second or third, which happens to 
be of the same kind with it ; then that extent will reach from the 
other term to the fourth term, as required, taking both extents 
towards the same end of the scale. 

In every triangle, or case in trigonometry, there must be given 
three parts, to find the other three. And, of the three parts that 
are given, one of them at least must be a side ; because the same 
angles are common to an infinite number of triangles. 

All the cases in trigonometry may be comprised in three vari- 
eties only ; viz. 

1. When a side and its opposite angle are given. 

2. When two sides and the contained angle are given. 

3. When the three sides are given. 

For there cannot possibly be more than these three varieties of 
cases ; for each of which it will therefore be proper to give a sepa- 
rate theorem, as follows : 

Wlien a side and its opposite angle are two of the given parts. 
Then the sides of the triangle have the same proportion to each 
other, as the sines of their opposite angles have. 
That is, 

As any one side. 

Is to the sine of its opposite angle ; 
So is any other side, 
To the sine of its opposite angle. 
For, let ABC be the proposed triangle, having 
AB the greatest side, and BC the least. Take 
AD = BC, considering it as a radius ; and let 
fall the perpendiculars DE, CF, which will evi- ^ 
dently be the sines of the angles A and B, to the radius AD or 
BC. But the triangles ADE, ACF, are equiangular, and there- 
fore AC ; CF : : AD or BC : DE ; that is, AC is to the sine of its 
opposite angle B, as BC to the sine of its opposite angle A. 

In practice, to find an angle, begin the proportion with a side 





364 THE PRACTICAL MODEL CALCULATOR. 

opposite a given angle. And to find a side, begin with an angle 
opposite a given side. 

An angle found by this rule is ambiguous, or uncertain whether 
it be acute or obtuse, unless it be a right angle, or' unless its mag- 
nitude be such as to prevent the ambiguity ; because the sine an- 
swers to two angles, which are supplements to each other ; and 
accordingly the geometrical construction forms two triangles with 
the same parts that are given, as in the example below ; and when 
there is no restriction or limitation included in the question, either 
of them may be taken. The degrees in the table, answering to the 
sine, are the acute angle ; but if the angle be obtuse, subtract those 
degrees from 180°, and the remainder will be the obtuse angle. 
When a given angle is obtuse, or a right one, there can be no am- 
biguity ; for then neither of the other angles can be obtuse, and 
the geometrical construction will form only one triangle. 

In the plane triangle ABC, 

fAB 345 yards 
Given, { BC 232 yards 
(angle A 37° 20' 
Required the other parts. i ^ 

G-eometrically. — Draw an indefinite line, upon which set off AB 
= 345, from some convenient scale of equal parts. Make the 
angle A = 37|-°. With a radius of 232, taken from the same 
scale of equal parts, and centre B, cross AC in the two points C, C. 
Lastly, join BC, BC, and the figure is constructed, which gives 
two triangles, showing that the case is ambiguous. 

Then, the sides AC measured by the scale of equal parts, and 
the angles B and C measured by the line of chords, or other in- 
strument, will be found to be nearly as below ; viz. 

AC 174 angle B 27° angle C 115J° 

or 374J or 78i or 64^ 

Arithmetically. — First, to find the angles at C : 

As side BC 232 log. 2-3654880 

To sin. opp. angle A 37° 20' 9-7827958 

So side AB345 2-5378191 

To sin. opp. angle C 115° 36' or 64° 24 9-9551269 

Add angle A 37 20 37 20 

The sum 152 56 or 101 44 

Taken from 180 00 180 00 

Leaves angle B 27 04 or 78 16 

Then, to find the side AC : 

As sine angle A 37° 20' log. 9-7827958 

To opposite side BC 232 2-365488 

c . 1 -o /27°04' 9-6580371 

So sine angle B | ^g ^^ 9-9908291 

To opposite side AC 174-07 2-2407293 

or, 374-56 2-5735213 



TRIGONOMETRY. 



365 



In the plane triangle ABC, 

r AB 365 poles 

Given, { angle A 57° 12' 

( angle B 24 45 Ans. 

Required the other parts. 

In the plane triangle ABC, 

r AC 120 feet 

Given, ^ BC 112 feet 

(angle A 57° 27' 
Required the other parts. Ans. 



{ 



angle C 98° 3' 
AC 154-33 
BC 309-86 



r angle B 64° 34' 21'' 

or, 115 25 39 

angle C 57 58 39 

1 or, 7 7 21 

I AB 112-65 feet 

(^ or, 16-47 feet 



When two sides and their contained angle are given. 
Then it will be. 
As the sum oft those two sides, 
Is to the difference of the same sides ; 
So is the tang, of half the sum of their opposite angles, 
To the tang, of half the difference of the same angles. 

Hence, because it is known that the half sum of any two quan- 
tities increased by their half difference, gives the greater, and di- 
minished by it gives the less, if the half difference of the angles, 
so found, be added to their half sum, it will give the greater angle, 
and subtracting it will leave the less angle. 

Then, all the angles being now known, the unknown side will be 
found by the former theorem. 

Let ABC be the proposed triangle, having 
the two given sides AC, BC, including the given 
angle C. With the centre C, and radius CA, 
the less of these two sides, describe a semicircle, 
meeting the other side BC produced in D and E. 
Join AE, AD, and draw DF parallel to AE. 

Then, BE is the sum, and BD the difference of the two given 
sides CB, CA. Also, the sum of the two angles CAB, CBxl, is 
equal to the sum of the two CAD, CDA, these sums being each 
the supplement of the vertical angle C to two right angles : but 
the two latter CAD, CDA, are equal to each other, being opposite to 
the two equal sides C A, CD : hence, either of them, as CDA, is equal 
to half the sum of the two unknown angles CAB, CBA. Again, 
the exterior angle CDA is equal to the two interior angles B and 
DAB ; therefore, the angle DAB is equal to the difference between 
CDA and B, or between CAD and B ; consequently, the same 
angle DAB is equal to half the difference of the unknown angles 
B and CAB ; of which it has been shown that CDA is the half sum. 

Now the angle DAE, in a semicircle, is a right angle, or AE is 
perpendicular to AD ; and DF, parallel to AE, is also perpendicular 
2f2 




366 THE PRACTICAL MODEL CALCULATOR. 

to AD : consequently, AE is the tangent of CDA the half sum^ 
and DF the tangent of DAB the half difference of the angles, to 
the same radius AD, by the definition of a tangent. But, the tan- 
gents AE, DE, being parallel, it will be as BE : BD : : AE : DE ; 
that is, as the sum of the sides is to the diifference of the sides, so 
is the tangent of half the sum of the opposite angles, to the tan- 
gent of half their difference. 

The sum of the unknown angles is found, by taking the given 
angle from 180°. 

In the plane triangle ABC, 

r AB 345 yards ^ 

Given, { AC 174-07 yards 

(angle A 37° 20' ^ b 

Required the other parts. 

G-eometrieally. — Draw AB = 345 from a scale of equal parts. 
Make the angle A = 37° 20'. Set off AC = 174 by the scale of 
equal parts. Join BC, and it is done. • 

Then the other parts being measured, they are found to be nearly 
as follows, viz. the side BC 232 yards, the angle B 27°, and the 
angle C 1151°. 

Arithmetieally. 

As sum of sides AB, AC 519-07 log. 2-7152259 

To difi'erence of sides AB, AC 170-93 2-2328183 

So tangent half sum angles C and B 71° 20' 10-4712979 

To tangent half difference angles C and B 44 16 9-9888903 

Their sum gives angle C 115 36 
Their diflf. gives angle B 27 4 

' Then, by -the former theorem, 

As sine angle C 115° 36', or 64° 24' log. 9-0551259 

To its opposite side AB 345 2-5378191 

So sine angle A 37° 20' 9-7827958 

To its opposite side BC 232 2-3654890 

In the plane triangle ABC, 

r AB 365 poles 

Given, ^ AC 154-33 

(angle A 57° 12' ( BC 309-86 

Required the other parts. < angle B 24° 45' 

(angle C 98° 3' 

In the plane triangle ABC, 

f AC 120 yards 

Given, ^ BC 112 yards 

( angle C 57° 58' 39" ( AB 112-65 

Required the other parts. \ angle A 57° 27' 0'' 

(angle B 64 34 21 



TRIGONOMETRY. 367 

When the three sides of the tjnangle are given. 

Then, having let fall a perpendicular from the greatest angle 
upon the opposite side, or base, dividing it into two segments, and 
the whole triangle into two right-angled triangles ; it will be, 

As the base, or sum of the segments, 

Is to the sum of the other two sides ; 

So is the difference of those sides, 

To the difference of the segments of the base. 

Then, half the difference of the segments being added to the 
half sum, or the half base, gives the greater segment; and the 
same subtracted gives the less segment. 

Hence, in each of the two right-angled triangles, there will be 
known two sides, and the angle opposite to one of them ; conse- 
quently, the other angles will be found by the first problem. 

The rectangle under the sum and difference of the two sides, is 
equal to the rectangle under the sum and difference of the two seg- 
ments. Therefore, by forming the sides of these rectangles into 
a proportion, it will appear that the sums and differences are pro- 
portional, as in this theorem. 

In the plane triangle ABC, 

fAB 345 yards 
Given, the sides ^ AC 232 

(BC 174-07 
To find the angles. 

G-eometrically. — Draw the base AB = 345 by a scale of equal 
parts. With radius 232, and centre A, describe an arc ; and with 
radius 174, and centre B, describe another arc, cutting the former 
in C. Join AC, BC, and it is done. 

Then, by measuring the angles, they will be found to be nearly 
as follows, viz. angle A 27°, angle B 37 J°, and angle C 115 J°. 

Arithmetically. — Having let fall the perpendicular CP, it will be, 
As the base AB : AC + BC : : AC — BC : AP - BP' 
that is, as 345 : 406-07 : : 57-93 : 68-18 = AP - BP 

its half is 34-09 

the half base is 172-50 

the sum of these is 206-59 = AP 

and their difference 138-41 = BP 

Then, in the triangle APC, right-angled at P, 

As the side AC 232 log. 2-3654880 

To sine opposite angle 90° 10-0000000 

So is side AP 206-59 2-3151093 

To sine opposite angle ACP 62° ^^' 9-9496213 

Which taken from 90 00 

Leaves the angle A 27 04 




368 



THE PRACTICAL MODEL CALCULATOR. 



Again, in the triangle BPC, right-angled at P, 

As the side of BC 174-07 log. 2-2407239 

To sine opposite angle P.. . 90° 10-0000000 

So is side BP 138-41 2-1411675 

To sin. opposite angle BCP 52° 40' 9-9004436 

. Which taken from 90 00 

Leaves the angle B... 37 20 

Also, the angle ACP... 62° 56' 

Added to angle BCP... 52 40 

Gives the whole angle ACB...115 36 

So that all the three angles are as follow, viz. 
the angle A 27° 4'; the angle B 37° 20'; the angle C 115° 36'. 

In the plane triangle ABC, 

f AB 365 poles 
Given the sides, { AC 154-33 

(BC 309-86 
To find the angles. 



In the plane triangle ABC, 
(AB 120 
Given the sides, ^ AC 112-65 
(BC 112 

To find the angles. 



angle A 57° 12' 
angle B 24 45 
angle C 98 3 



Tangle A 57' 
< angle B 57 
( angle C 64 



27' 00" 
58 39 
34 21 



The three foregoing theorems include all the cases of plane tri- 
angles, both right-angled and oblique ; besides which, there are 
other theorems suited to some particular forms of triangles, which 
are sometimes more expeditious in their use than the general ones ; 
one of which, as the case for which it serves so frequently occurs, 
may be here taken, as follows : — 

When^ in a right-angled triangle, there are given one leg and the 
angles ; to find the other leg or the hypothenuse ; it will be, 

As radius, i. e. sine of 90° or tangent of 45° 

Is to the given leg. 

So is the tangent of its adjacent angle 

To the other leg ; 

And so is the secant of the same angle 

To the hypothenuse. 

AB being the given leg, in the right-angled tri- 
angle ABC ; with the centre A, and any assumed ra- 
dius, AD, describe an arc DE, and draw BF perpen- 
dicular to AB, or parallel to BC. Now it is evident, 
from the definitions, that DF is the tangent, and AF 
the secant, of the arc DE, or of the angle A which 
is measured by that arc, to the radius AD. Then, because of the 
parallels BC, DF, it will be as AD : AB :: DF : BC : : AF : AC, 
which is the same as the theorem is in words. 




OF HEiaHTS AND DISTANCES. 



369 



to find AC andBC. 



.log. 



In the right-angled triangle ABC, ' 
^. /the leg AB 162 1 
^^^^^ i angle A 53° r 48'' I 
Greometrically. — Make AB = 162 equal parts, and the angle A = 
53° 7' 48" ; then raise the perpendicular BC, meeting AC in C. 
So shall AC measure 270, and BC 216. 
Arithmetically, 

As radius tang. 45° . 

TolegAB 162 . 

So tang, angle A 53° 7' 48'' . 

TolegBC 216 . 

So secant angle A 53° 7' 48". 

To hyp. AC 270 . 

In the right-angled triangle ABC, 

Given 

To find the other two sides. 



10-0000000 
2-2095150 

10-1249371 
2-3344521 

10-2218477 
2-4313627 



r the leg AB 180 

t the angle A 62° 40' J AC 392-0147 

\ BC 348-2464 

There is sometimes given another method for right-angled tri- 
angles, which is this : 

ABC being such a triangle, make one leg AB ra- 
dius, that is, with centre A, and distance AB, de- 
scribe an arc BF. Then it is evident that the other 
leg BC represents the tangent, and the hypothenuse 
AC the secant, of the arc BF, or of the angle A. 

In like manner, if the leg BC be made radius ; 
then the other leg AB will represent the tangent, and 
thenuse AC the secant, of the arc BG or angle C. 

But if the hypothenuse be made radius ; then each leg Avill re- 
present the sine of its opposite angle ; namely, the leg AB the sine 
of the arc AE or angle C, and the leg BC the sine of the arc CD 
or angle A. 

And then the general rule for all these cases is this, namely, 
that the sides of the triangle bear to each other the same propor- 
tion as the parts which they represent. 

And this is called. Making every side radius. 




hypo- 



OF HEIGHTS AND DISTANCES. 

By the mensuration and protraction of lines and angles, are de- 
termined the lengths, heights, depths, and distances of bodies or 
objects. 

Accessible lines are measured by applying to them some certain 
measure a number of times, as an inch, or foot, or yard. But in- 
accessible lines must be measured by taking angles, or by some 
such method, drawn from the principles of geometry. 

When instruments are used for taking the magnitude of the 

24 _ 



370 THE PRACTICAL MODEL CALCULATOR. 

angles in degrees, the lines are then calculated by trigonometry : 
in the other methods, the lines are calculated from the princi- 
ple of similar triangles, without regard to the measure of the 
angles. 

Angles of elevation, or of depression, are usually taken either 
with a theodolite, or with a quadrant, divided into degrees and mi- 
nutes, and furnished with a plummet suspended from the centre, 
and two sides fixed on one of the radii, or else with telescopic 
sights. 

To take an angle of altitude and depression with the quadrant. 

Let A be any object, as the sun, a 

moon, or a star, or the top of a tower, ^/' 

or hill, or other eminence ; and let it / 

be required to find the measure of the ^/ 

angle ABC, which a line drawn from y^* 

the object makes with the horizontal 
line BC. 

Fix the centre of the quadrant in 
the angular point, and move it round 
there as a centre, till with one eye at 
D, the other being shut, you perceive the object A through the 
sights : then will the arc GH of the quadrant, cut ofi" by the plumb 
line BH, be the measure of the angle ABC, as required. 




The angle ABC of depression of any ob- g/\ 

ject A, is taken in the same manner; except ''^^- ^\ 

that here the eye is applied to the centre, and j, \^ 
the measure of the angle is the arc GH, on \ 

the other side of the plumb line. \ 

'^\ A 

The following examples are to be constructed and calculated by 
the foregoing methods, treated of in trigonometry. 

Having measured a distance of 200 feet, in a direct horizontal 
line, from the bottom of a steeple, the angle of elevation of its top, 
taken at that distance, was found to be 47° 30' : from hence it is 
required to find the height of the steeple. 

Construction. — Draw an indefinite line, upon which set off AC = 
200 equal parts, for the measured distance. Erect the indefinite 
perpendicular AB ; and draw CB so as to make the angle C =»* 
47° 30', the angle of elevation; and it is done. Then AB, mea- 
sured on the scale of equal parts, is nearly 218i. 

Calculation. 

As radius 10-0000000 

To AC 200 2-3010300 

So tang, angle C 47° 30' 10-0379475 / 

To AB 218-26 required 2-3389775 ^ 




OF HEiaHTS AND DISTANCES. 371 

What was the perpendicular height of a cloud, or of a balloon, 
when its angles of elevation were 35° and 64°, as taken by two 
observers, at the same time, both on the same side of it, and in 
the same vertical plane ; their distance, as under, being half a mile, 
or 880 yards. And what was its distance from the said two ob- 
servers ? 

Construction. — Draw an indefinite ground line, upon which set 
off the given distance AB = 880 ; then A and B are the places 
of the observers. Make the angle A = 35°, and the angle B = 
64° ; and the intersection of the lines at C will be the place of the 
balloon ; from whence the perpendicular CD, being let fall, will be 
its perpendicular height. Then, by measurement, are found the 
distances and height nearly, as follows, viz. AC 1631, BC 1041, 
DC 936. c 

Calculatiojab, 

First, from angle B 64° ^^,-'''''/ 

Take angle A 35 ,,-''' / 

Leaves angle ACB 29 ,,-''* 



Then, in the triangle ABC, a b d 

As sine angle ACB 29° 9-6855712 

To opposite side AB 880 2-9444827 

So sine angle A 35° 9-7585913 

To opposite side BC 1041-125 3-0175028 

As sine angle ACB 29° -9-6855712 

To opposite side AB 880 2-9444827 

So sine angle B 116°or64° 9-9536602 

To opposite side AC 1631-442 3-2125717 

And, in the triangle BCD, 

As sine angle D 90° 10-0000000 

To opposite side BC 1041-125 3-0175028 

So sine angle B 64° 9-9536602 

To opposite side CD 935-757 2-9711630 

Having to find the height of an obelisk standing on the top of a 
declivity, I first measured from its bottom, a distance of 40 feet, 
and there found the angle, formed by the oblique plane and a line 
imagined to go to top of the obelisk 41° ; but, after measuring on 
in the same direction 60 feet farther, the like angle was only 23° 45'. 
What then was the height of the obelisk ? 

Construction. — Draw an indefinite line for the sloping plane or 
declivity, in which assume any point A for the bottom of the 
obelisk, from whence set off the distance AC = 40, and again 
CD = 60 equal parts. Then make the angle C =41°, and the 
angle D = 23° 45' ; and the point B, where the two lines meet, 
will be the top of the obelisk. Therefore AB, joined, will be its 
height. 



372 



THE PRACTICAL MODEL CALCULATOR. 




Calculation. 
From the angle C 41° 00' 
Take the angle D 23 45 
Leaves the angle DBC 17 15 



Then, in the triangle DBC, 

As sine angle DBG 17° W 9-4720856 

To opposite side DC 60 1*7781513 

So sine angle D . 24 45 9-6050320 

To opposite side CB 81-488 1-9110977 

And, in the triangle ABC, 

As sum of sides CB, CA 121-488 2-0845333 

To difference of sides CB, CA 41-488 1-6179225 

So tang, half sum angles A, B 69°30' 10-4272623 

42 241 9-9606516 



To tang, half diff. angles A, B 



5i 



The diff. of these is angle CBA 27 

Lastly, as sine angle CBA 27° 5J' 9-6582842 

To opposite side CA 40 1-6020600 

So sine angleC 41°0' 9-8169429 

To opposite side AB -57-623 1-7607187 

"Wanting to know the distance between two inaccessible trees, or 
other objects, from the top of a tower, 120 feet high, which lay in 
the same right line with the two objects, I took the angles formed 
by the perpendicular wall and lines conceived to be drawn from the 
top of the tower to the bottom of each tree, and found them to be 
33° and 64J°. What then may be the distance between the two 
objects ? A 

Construction. — Draw the indefinite 
ground line BD, and perpendicular to 
it BA = 120 equal parts. Then draw 
the two lines AC, AD, making the two 
angles BAC, BAD, equal to the given 
angles 33° and 64|°. So shall C and D be the places of the two 
objects. 

Calculation. — First, In the right-angled triangle ABC, 

As radius ..10-0000000 

ToAB - 120 2-0791812 

So tang, angle BAC 33° 9-8125174 

ToBC 77-929 1-8916986 

And, in the right-angled triangle ABD, 

As radius 10-0000000 

To AB 120 2-0791812 

So tang, angle BAD 

ToBD 251-585 

From which take BC 77-929 

Leaves the dist. CD 173-656 as required. 




64}° 10-3215039 

2-4006851 



SPHERICAL TRIGONOMETRY. 373 

Being on the side of a river, and wanting to know the distance 
to a house which was seen on the other side, I measured 200 yards 
in a straight line by the side of the river ; and then at each end 
of this line of distance, took the horizontal angle formed between 
the house and the other end of the line ; which angles were, the 
one of them 68° 2', and the other 73° 15'. What then were the 
distances from each end to the house ? 

Construction.— Dvd^vf the line AB = 200 equal parts. Then 
draw AC so as to make the angle A = 68° 2', and BC to make 
the angle B = 73° 15'. So shall the point C be the place of the 
house required. ^_ 

Calculation. — ^^i| 

To the given angle A 68° 2' ^^ ~^ 

Add the given angle B 73 15 

Then their sum 141 17 

Being taken from 180 / 

Leaves the third angle C 38 43 ,^?^^^.;g^^^^^'* 

Hence, As sin. angle C 38° 43' 9-7962062^ 

To op. side AB 200 2-3010300 

So sin. angle A 68° 2' 9-9672679 

To op. side BC 296-54 2-4720917 

And, As sin. angle C 38° 43' ; 9-7962062 

To op. side AB 200 2-3010300 

So sin. angle B 73° 15' 9-9811711 

To op. side AC 306-19 2-4859949 




SPHERICAL TRIGONOMETRY. 

This Article is taken from a short Practical Treatise on Spherical Trigonometry, 
by Oliver Byrne, the author of tlie present icork. Published by J. A. Valpy. 
London, 1835. 

As the sides and angles of spherical triangles are measured by 
circular arcs, and as these arcs are often greater than 90°, it may 
be necessary to mention one or two particulars respecting them. 

The arc CB, which when added to 
AB makes up a quadrant or 90°, is 
called the complement of the arc AB ; 
every arc will have a complement, 
even those which are themselves 
greater than 90°, provided we con- 
sider the arcs measured iii the direc- 
tion ABCD, &c., as positive, and 
consequently those measured in the 
opposite direction as negative. The 
complement BC of the arc AB com- 
mences at B, where AB terminates, 
and may be considered as generated by the motion of B, the ex- 
2G 




874 THE PRACTICAL MODEL CALCULATOR. 

tremity of the radius OB, in the direction BC. But the comple- 
ment of the arc AD or DC, commencing in like manner at the ex- 
tremity D, must be generated by the motion of D in the opposite 
direction, and the angular magnitude AOD will here be diminished 
by the motion of OD, in generating the complement ; therefore 
the complement of AOD or of AD may with propriety be consi- 
dered negative. 

Calling the arc AB or AD, &, the complement will be 90° — ; 
the complement of 36° 44' 33'' is 53° 15' 21" ; and the complement 
of 136° 2T' 39'' is negative 46° 27' 39''. 

The arc BE, which must be added to AB to make up a semi- 
circle or 180°, is called the §u2?plement of the arc AB. If the arc 
is greater than 180°, as the arc ADF its supplement, FE mea- 
sured in the reverse direction is negative. The expression for the 
supplement of any arc o is therefore 180° — e ; thus the supple- 
ment of 112° 29' 35" is 67° 30' 25", and the supplement of 205° 
42' is negative 25° 42'. 

In the same manner as the complementary and supplementary 
arcs are considered as positive or negative, according to the di- 
rection in which they are measured, so are the arcs themselves 
positive or negative ; thus, still taking A for the commencement, 
or origin^ of the arcs, as AB is positive, AH will be negative. In 
the doctrine of triangles, we consider only positive angles or arcs, 
and the magnitudes of these are comprised between 9 = and o = 
180° ; but in the general theory of angular quantity, we consider 
both positive and negative angles, according as they are situated 
above or below the fixed line AO, from which they are measured, 
that is, according as the arcs by which they are estimated are posi- 
tive or negative. Thus the angle BOA is positive, and the angle 
AOH negative. Moreover, in this more extended theory of angular 
magnitude, an angle may consist of any number of degrees what-x 
ever ; thus, if the revolving line OB set out from the fixed line A, 
and make n revolutions and a part, the angular magnitude gene- 
rated is measured by n times 360°, plus the degrees in the ad- 
ditional part. 

In a right-angled spherical triangle we are to recognise but five 




parts, namely, the three sides «, 5, <?, and the two angles A, B ; 



so that the right angle C is omitted. 



SPHERICAL TRIGONOMETRY. 



375 




Let A', c\ B/ be the comple- 
ments of A, c, B, respectively, 
and suppose 5, <z, B', c' ^ A', to be 
placed on the hand, as in the 
annexed figure, and that the 
fingers stand in a circular order, 
the parts represented by the 
fingers thus placed are called 
circular parts. 

If we take any one of these as 
a middle part, the two which lie 
next to it, one on each side, will 
be adjacent parts. The two parts 
immediately beyond the adjacent 
parts, one on each side, are called 
the opposite parts. 

Thus, taking A' for a middle part, h and g' will be adjacent parts, 
and a and B' are opposite parts. 

If we take o' as a middle part, A' and B' are adjacent parts, and 
J, «, opposite parts. 

When B' is a middle part, c', «, become adjacent parts, and A', 
5, opposite parts. 

Again, if we take a as a middle part, then B', 5, will be adjacent 
parts, and c\ A', opposite parts. 

Lastly, taking 5 as a middle part, A', a, are adjacent parts, and 
c', B', opposite parts. 

This being understood, Napier's two rules may be expressed as 
follows : — 

I. Bad. X sin. middle part = product of tan. adjacent parts. 

II. Bad. X sin. middle part = product of cos. opposite parts. 
Both these rules maybe comprehended in a single expression, thus. 

Bad. sin. mid. = prod. tan. adja. = prod. cos. opp. ; 
and to retain this in the memory we have only to remember, that 
the vowels in the contractions sm., tan.^ cos.^ are the same as those 
in the contractions mid., adja., opp., to which they are joined. 

These rules comprehend all the succeeding equations, reading 
from the centre, B = radius. 

In the solution of right-angled spherical triangles, two parts are 
given to find a third, therefore it is necessary, in the application of 
this formula, to choose for the middle part that which causes the 
other two to become either adjacent parts or opposite parts. 
In a right-angled spherical triangle, the hypothenuse 
c = 61° 4' 56'^ ; and the angle 
A =. 61° 50' 29". Required the adjacent leg ? 



90° 

= 61 



0' 
4 



00" 

56 



28 55 04 = C. 



90° 

A = 61 



0' 
50 



00" 

29 



28 \) 31 = A. 



376 



THE PKACIICAL MODEL CALCULATOR. 




.A' 



^ 









•o- 



•i 



#. 



•soa ■nsoQ — '/ 



% 



\^Tan. h .tan . c\ % 



P 



V, 



k 






In this example, A' is selected for the middle part, because then 
h and c' become adjacent parts, as in the annexed figure. 

E,ad. X sin. A' = tan. h X tan. e' 

rad. X sin. A' 

.*. tan. = 




tan c' 

By Logarithms. 

Rad. - -10-0000000 

Sin. A^-28°9^21^^ - 9-6738628 

19-6738628 
Tan.c'-28°55'4''- 9-7422808 

Tan.y-40°30a6''-^9315820 
The side adjacent to the given 
angle is acute or obtuse, accord- 
ing as the hypothenuse is of the 
same, or of different species with the given angle. 
.-. the leg h = 40° 30' 16'', acute. 
Supposing the hypothenuse c = 113° 55', and the angle A = 31° 51', 
then the adjacent leg h would be 117° 34', obtuse. 



SPHERICAL TRIGONOMETRY. 



37T 



In tlie right-angled spherical triangle ABC, are given the hypo- 
thenuse c = 113° 55', and the angle A = 104° 08'; to find the 
opposite leg a. 



c = 113° 55' 
90 




23 55 = c'. 

A = 104° 08' 
90 

14 08 = A' 



In this example, a is taken for the middle part, then A' and 
are opposite parts. (See the subjoined figure.) 

From the general formula, we 
have. 

Had. X sin. a = cos. A' x cos. c'. 

COS. A' X COS. e' 

.*. sm. a = =f^— 5 . 

Rad. 

By Logarithms. 

COS. A' - 14° 08' 9-9860509 

COS. c' -23 55 9-9610108 



sm. a 



19-9476617 

Radius 10-0000000 

/ 117° 34' \ 

t 62 26 ]'" 



9-9476617 




The obtuse side 117° 34' is the leg required, for the side oppo- 
site to the given angle is always of the same species with the 
given angle. 

If in a right-angled spherical 
triangle, the hypothenuse were 
78° 20', and the angle A = 
37° 25', then the opposite leg 
a = 36° 31', and not 143° 29', 



because the given angle is acute. 
In a right-angled spherical tri- 
angle, are given c = 78° 20', and 
A = 37° 25', to find the angle B. 


90° 


0' 


c = 78 


20 


11 


40 = c'. 


90° 


0' 


A = 37 


•25 


52 


35= A' 


2 G 2 






378 



THE PRACTICAL MODEL CALCULATOR. 



Here the complement of tlie hypothenuse (c^) is the middle part; 
and the complement of the ^^-—^ V 

angle opposite the perpen- 
dicular (A^), and the com- 
plement of the angle oppo- 
site the base (B') are the 
adjacent parts. This will 
readily be perceived by 
reference to the usual 
figure in the margin. 

Had. X sin. c' — tan. A' 
X tan. B' ; 

Rad. X sin. c' 
•■•'""•^= tan. A' • 
By Logarithms. 

Rad 10-0000000 

sin. e' - 11° 40^ 9-3058189 
19-3058189 
tan. A' - 52° 35^ 10-1163279 
.-.tan. B'- 8° 48' 9-1894910 
But 90 - B = B' 
hence 90 - B' = B. 
90° 0' 
8 48 
B = 81° 12'. 

When the hypothenuse and an angle are given, the other angle is 
acute or obtuse, according as the given parts are of the same or of 
different species. 

In the above example, both the given parts are acute, therefore 
the required angle is acute ; but if one be acute and the other ob- 
tuse, then the angle found would be obtuse : — Thus, if the hypo- 
thenuse be 113° bb', and the angle A = 31° 51' ; then will B' = 
14° 08', and the angle B = 104° 08'. 

Given the hypothenuse c = 61° 04' b&'^ and the side or leg, 
a = 40° 30' 20", to find the angle adjacent to a. c' 

90° 0' 0" 
c = 61 04 56 

28 55 04" = (?". 

The three parts are here 
connected ; therefore the com- 
plement of B is the middle 
part, a and the complement of 
c are the adjacent parts. 

Hence we have, 
Rad. X sin. B' = tan. a X tan. c'. 
tan. a X tan. c' 

••• ^^-^^ s = — Eai — 





SPHERICAL TRIGONOMETRY. 



379 




By Logarithms. 

tan. a - 40° 30' 20^' = 9-9315841 
tan. o' - '28 bb 04 = 9-7422801 

19-6738642 
Ead... 10-0000000 

sin. B'....28° 09' 31'' 9-6738642 

90° 0' 0" 
B' = 28 09 31 

61 50 29 = B. 

The angle adjacent to the given side is acute or obtuse accord- 
ing as the hypothenuse is of the same or of different species with 
the' given side. 

Before working the above example, it was easy to foresee that 
the angle B would be acute ; but suppose the hypothenuse = 70° 
20', and the side a = 117° 34', then the angle B would be obtuse, 
because a and e are of different species. 

Rule V. — In a spherical triangle, right-angled at c, are given 
c = 78° 20' and h = 117° 34', to find the angle B ; opposite the 
given leg, (see the next diagram.) 

In this example, b becomes the middle part, and g' and B' oppo- 
site parts ; and therefore, by the rule, 

Bad. X sin. h = cos. B' X cos. c' ; that is, 

^ Rad. X sin. h ^^-"^^ ^^' 

cos. B' = -, . 

COS. c' 

90° ~ 78° 20' = 11° 40' = o'. // 3. 

Hence, by Logarithms. 

Rad 10-0000000 / "^^J ^^^ 

sin. b = sin. 117° 34' \ 9.947^^.^ 

or sin. 62 26 / ^ ^47bb55 ^ 

19-9476655 f'^"^^. ' 
cos. c 11° 40' 9-9909338 / ^ ' 

COS. B' 25° 09' 9-9567317 ' ^ 




380 



THE PRACTICAL MODEL CALCULATOR. 
B 




But since the angle 
opposite the given 
side is of the same 
species with the given 
side, 90° must be 
added to B', to pro- 
duce B :— viz. 90° + 
25° 09^ = 115° 09'. 

Given c = 61° 04' 
bQ'\ and h = 40° 30' 
20", to find the other 
side a. 

Here c' is the mid- 
dle part, a and h the 
opposite parts ; hence 
by position 4, a = 50° 30' 30". 

Given the side 5 = 48° 24' 16", and the adjacent angle A = 
660 20/ 40", to find the side a. 

In this instance, h is the middle part, the complement of A and 
a are adjacent parts. Consequently, a = 59° 38' 27". 

In the right-angled spherical triangle ABC, 

^. r The side a = 59° 38' 27" \ ^ xx a ^v i a 

^^^^^i Its adjacent angle B = 52° 32' 55" |*^ ^^^ *^^ ^^S^^ -^• 

Answer, m"" 20' 40". 

The required angle is of the same species as the given side, and 
vice versa. 

Given the side h = 49° 17', and its adjacent angle A = 23° 28', 
to find the hypothenuse. 

Making A' the middle part, the others will be adjacent parts, 
and, therefore, by the first rule we have (? = 51° 42' 37". 

In a spherical triangle, right-angled at C, are given b = 29° 12' 
50", and B = 37° 26' 21", to find the side a. 

Taking a for the middle part, the other two will be adjacent parts ; 
hence by the rule. 

Bad. X sin. a = tan. h X tan. B' 

that is, rad. X sin. a = tan. b X cot. B 

tan. b X cot. B 

.'. sm. a = T 

rad. 

In this case, there are two solutions, i. e. a and the supple- 
ment of a, because both of them have the same sine. As sin. a 
is necessarily positive, b and B must necessarily be always of 
the same species, so that, as observed before, the sides including 
the right angle are always of the same species as the opposite 
angles. 



SPHERICAL TRIGONOMETRY. 



381 




In working this example, 
we find the log. sin. a = 
9-8635411, which corre- 
sponds to 46° 55' 02'', 
or, 133° 04' 58". 

It appears, therefore, 
that a is ambiguous, for 
there exist two right-angled 
triangles, having an oblique 
angle, and the opposite side 
in the one equal to an 
oblique angle and an oppo- 
site side in the other, but 
the remaining oblique angle 
in the one the supplement 
of the remaining oblique 
angle in the other. These triangles are situated with respect 
to each other, on the sphere, as the triangles ABC, ADC, 
in the annexed diagram, in which, with the exception of the 
common side AC, and the equal angles B, D, the parts of the 
one triangle are supplements of the corresponding parts of the 
other. 

In a right-angled spherical triangle are 

p. ("the side a = 42° 12', 1 to find the adjacent 

^^^^^ \ its opposite angle A = 48° J angle B. 

The complement of the given angle is the middle part; and 
neither a nor B' being joined to A', they are consequently opposite 
parts ; hence, the angle B = 64° 35', or 115° 25' ; this case, like 
the last, being ambiguous, or doubtful. 

Given a = 11° 30', and A = 23° 30', to find the hypothenuse c. 
c = 30°, or 150°, being ambiguous. 

In a right-angled triangle, there are given the two perpendicu- 
lar sides, viz. a = 48""" 



angle A. 



24' 16", b = 59° 3^ 



A = 66° 20' 40". 



27", to find the 



Given a = 142° 31', b = 54° 22', to find c. 
c = 117° 33'. 

r A = 37° 25' 1 
Given ^ t^ _ qi i o i ^^^^^^^^^ *^^ si^® ^• 

a = 36° 31'. 



C A _ ggO OQf AQf/ 1 

Given ^ g ^ 50 39 55 r^ ^^^ *^® hypothenuse c. 
c = 70° 23' 42". 



382 



THE PRACTICAL MODEL CALCULATOR. 



MEASUREMENT OF ANGLES. 

Fi'om the " Civil Engineer and Architect's Journal," for Oct. and Nov. 1847. 

A NEW METHOD OP MEASURING THE DEGREES, MINUTES, ETC., IN ANY 
RECTILINEAR ANGLE BY COMPASSES ONLY, WITHOUT USING SCALE OR 
PROTRACTOR. 

Apply AB = x, from B to 1 ; from 1 to 2 ; from 2 to 3 ; from 
3 to 4 ; from 4 to 5. Then take B 5, in the compasses, and apply 
it from B to 6 ; from 6 to 7 ; from 7 to 8 ; from 8 to 9 ; and from 
9 to 10, near the middle of the arc AB. With the same opening, 




B 5 or A 4, or y, which we shall term it, lay off 4,11, 11,12, and 
12,13. Then the arc between 13 and 10 is found to be contained 
23 times in the arc AB. 



MEASUREMENT OP ANGLES. 



383 



Hence, we have, 



^x- y = 360°; 
^y -^ z = x; 
23 g = x: 01% 



X 

^ X 22 a; 

.-. 9^ + 23 =^, .•.^= 20f- 

By substituting this value in the first equation, we obtain, 
5 a; — 20^ = 360. 

1013 a; ^^^ , 360 X 207 ^„^ ^„, ^^ 

"207- = 360, and x = — ^^j^ — = 73° 33'-82. 

Apply AB == X, from B to 1 ; from 1 to 2 ; from 2 to 3 ; from 
3 to 4. Then take B 4, in the compasses, and apply it on the arc, 
from B to 4 ; from 4 to 5 ; from 5 to 6 ; from 6 to 7 ; and from 
7 to 8, near the middle of the arc AB. With the same opening, 
B4 = t/, lay off A 9, 9,10, 10,11, 11,12, 12,13, and 13,14. The 
arc between 14 and 8 is found to be contained nearly 24 times in 
the arc AB. Therefore, we have, 

^x + y = 360; 
lly— z = x\ 



^ ~ 24* 



24 2 = x\ or, 
a; 25 a; 

Substituting this value of y in the first equation, 



25a; 
4^ + 264 



360; 



360 X 264 
1071 



= 88° 44'-333. 



How to lay off an angle of any number of degrees^ minutes^ ^c, 
ivith compasses only^ without the use of scale or protractor. 
Let it be required 
to lay off an angle of 
36° 40' = /3. Take any 
small opening of the 
compasses less than 
one-tenth of the ra- 
dius, and lay off any 
number of equal small 
arcs, from A to 1 ; 
from 1 to 2 ; from 2 to 
3, &c., until we have 
laid off an arc, AB, 
greater than the one 
required. Draw B h 
through the centre 0, 
then will the arc a 6 = 
arc AB, which we shall 




384 



THE PRACTICAL MODEL CALCULATOR. 



put = 20 4) in this example, and proceed to measure a 5 as in the 
first example. Lay off a 5 from hto c; from c to d; from d to e; 
from e to/; from /to g. Putting g a = Ai, then, 

108 
6 X 20t + Ai = 360° = -^/3; because, 

360° 21600 108 
36° 40' ~ 2200 ~ 11 • 
Lay off, as before directed, g a, = Ai? from a to 7i, from h to s, 
and h to t\ then calling s t, Aa^ we have 

3 A, + A3 = 204'; 
and we find that s ^ is contained 28 times in the arc a h ; 



108 
... 120t+ A, = -3j^; 3 A, + A, 



!0 ^ ; and 28 A. = 20 ?. 



Eliminating Ai and As, we find 

29205 
3 = cyor^ ^ — 12'^ times $ nearly ; 

.-. 36° 40' = Z, A N is laid off with as much ease and certainty 
as by a protractor. 

As a second example, let it be required to lay off an angle of 
132° 2T'. From 180° 0' take 132° 27' = 47° 33', which put = ,J 
360° 2400 V V ^^^^ 

^f-o gg/ = -g-iy when put = ■, then I? = 360° = Tt, 

29 




We have laid off 29 small arcs from A to B ; 29 = ?. AB = 

ah = bc = cd = de=^Gf. And ag = hh = af = ^j^; hg = /\„. 

.-. 5 X 29t + Ai = 360° = ^i3 = w^e t ± A^ (1) 
2 A. - A, = 29t, or n A, ± Aa = ^t (2) 



13 A, 



29 



or 



^A. 



£t 



(3j 



MEASUREMENT OF ANGLES. 385 



Eliminating Aj an3 /\^, we have 

mnq ± {q^ l)}^"s ^ _ {5-2-13 + (13 + 1)}29'317 
nq 



^ - '■'-- ^~ 2400-2'13 



1323729 ^^^ . , „ ,. V XT :, 

~ 624Q0 ^ ~ i times 9 very nearly. Hence the line N deter- 
mines the angle a N = 132° 27'. 
In the expression 

vuq ^ ^ ' 

substituting the numerals of the first example, then 

{6-3-28 + (28 - 1)}20-11 29205 
3 = ^- j^Qg.3.2g ^ = "2268" * = 12'^ *i™^s 4> nearly, 

the result before obtained. 

The ambiguous signs of (E) cannot be mistaken or lead to error, 
if the manner in which it is deduced from (1), (2), (3), be attended 
to. From (3) 

A3 = — ; substituting this value of As, in (2), 

£ t 

wAi=f^qFAa = «^=F — ; which, when substituted for Ai 
in (1), gives 

V 1 / £ 4)\ 

-8 = m s^ ± - ys^ zip — J ; from which (R) is found. 

This method of measuring angles is more exact than it ^ay ap- 
pear ; for if, in the first example, we take 

5x — y = 360 ; 9y -j- z = x ; and 20 ;2 = a:, 
64800 
then X = -gg]- = 73° 33' 85. 

The first equations gave 73° 33' 82 when 23 ^ = x, so it does 
not matter much whether 20, 21, 22, 23, 24, or 25 times z make 
X. This fact is particularly worth attention. 

Given the three angles to find the three sides. 

The following formulas give any side a of any spherical triangle. 

. — COS. i S cos. (1 S — A) , 

sm. J a = v/ sinTB'sin.T; ' ^^^ 

^ cos. (I S - B) COS. (IS- C) 
^°^- i ^ = ^ sin. B sin. C. 

Given the three sides to find the three angles. 



sin . (I S - 5 ) s in, (j- S - c ) 

sin. b sin. c. 

sin. J S sin. (J S — a) 



8in. i A = ^/ ^i^_ J ^;^_ ^^ 



COS. I A = v/ • 7 • 

^ sm. sin. c. 



386 



GRAVITY-WEIGHT-MASS. 

ENTRE OF GRAVITY, AND OTHER C 

WEIGHTS OF ENGINEERING AND MECHANICAL MATERIALS. BRASS, 

COPPER, STEEL, IRON, WATER, STONE, LEAD, TIN, ROUND, SQUARE, FLAT, 
ANGULAR, ETC. 

1. In a second, the acceleration of a body falling freely in vacuo 
is 32*2 feet; what velocity has it acquired at the end of 5 seconds ? 

32-2 X 5 = 161 feet, the velocity. 

2. A cylinder rolling down an inclined plane with an initial velo- 
city of 24 feet a second, and suppose it to acquire each second 5 ad- 
ditional feet velocity ; what is its velocity at the end of 3*7 seconds ? 

24 + 3-7 X 5 = 42-5 feet. 

3. Suppose a locomotive, moving at the rate of 30 feet a second, 
(as it is usually termed, with a 30 feet velocity,) and suppose it to lose 
5 feet velocity every second ; what is its velocity at the end of 3-33 
seconds ? 

The acceleration is — 3-33, negative. 

.-. 30 - 5 X 3-33 = 13-35 feet. 

4. If a body has acquired a velocity of 36 feet in 11 seconds, 
by uniformly accelerated motion ; what is the space described ? 

36 X 11 

• 2 = 1^^ f^^*- 

5. A carriage at rest moves with an accelerated motion over a 
space of 200 feet in 45 seconds; at what velocity does it proceed 
at the beginning of the 46th second ? 

200 X 2 

— -— — = 8*8889 feet, the velocity at the end of the 45th second. 

The four fundamental formulas of uniformly accelerated motion are 
vt _pf _ v^ 

V the velocity, p the acceleration, t the time, and s the space. 

6. What space will a body describe that moves with an accele- 
ration of 11*5 feet for 10 seconds. 

11-5 X (loy 

2 

7. A body commences to move with an acceleration of 5-5 feet, 
and moves on until it is moving at the rate of 100 feet a second ; 
what space has it described ? 

(lOOf 



\2 

== 575 feet. 



2 X 5-5 



909-09 feet. 



= c -\- pt; s = ct + -2; s = — 2 — ^' ^~ 2 



GRAVITY — WEIGHT — MASS. 387 

8. A body is propelled with an initial velocity of 3 feet, and with 
an acceleration of 8 feet a second; what space is described in 
13 seconds ? 

3 X 13 -f- ^—^ = 715 feet. 

9. What distance will a body perform in 35 seconds, commenc- 
ing with a velocity of 10 feet, and being accelerated to move with 
a velocity of 40 feet at the beginning of the 36th second ? 

10+40 

2 X 35 = 875 feet, the distance. 

The formulas for a uniformly accelerated motion, commencing 
with a velocity c, are as follow : — 

pt^ c -\- V ^ v^ — c^ 

V 

The succeeding formulas are applicable for a uniformly retarded 
motion with an initial velocity g. 

pf' e -{■ V " c^ — v^ 

V = c — pt\ s = ct — -Y', « = ~2 — ^5 « = 2p ' 

10. A body rolls up an inclined plane, with an initial velocity 
of 50 feet, and suffers a retardation of 10 feet the second ; to what 
height will it ascend ? 

50 

vq = 5 seconds, the time. 

(50)2 
Q ^ -./^ = 125 feet, the height required. 

The free vertical descent of bodies in vacuo offers an important 
example of uniformly accelerated motion. The acceleration in the 
previous examples was designated by p, but in the particular mo- 
tion, brought about by the force of gravity, the acceleration is 
designated by the letter g^ and has the mean value of 32-2 feet. 

If this value of g be substituted for p^ in the preceding formula, 
we have, 

v = 32-2xf; t; = 8-024964 X >A; s^lQ'lxf; s= -015528 xv^-, 
t = -031056 X ?; ; and ^ = -2492224 x s/s. 

11. What velocity will a body acquire at the end of 5 seconds, 
in its free descent ? 

32-2 X 5 = 161 feet. 

12. What velocity will a body acquire, after a free descent 
through a space of 400 feet ? 

8-024964 X ^/400 = 160-49928 feet. 

13. What space will a body pass over in its free descent during 
10 seconds ? 

16-1 X (10)2 = 1610 feet. 



388 THE PRACTICAL MODEL CALCULATOR. 

14. A body falling freely in vacuo, lias in its free descent 
acquired a velocity of 112 feet ; what space is passed over ? 

•015528 X (112)2 ^ 194-783232 feet. 

15. In wliat time will a body falling freely acquire the velocity 
of 30 feet? 

•031056 X 30 = -93168 seconds. 

16. In what time will a body pass over a space of 16 feet, fall- 
ing freely in vacuo ? 

•2492224 x ^/16 = ^9968896 seconds. 

If the free descent of bodies go on, with an initial velocity, 
which we may call e, the formulas are, 

y = c + gt; v = c + S2'2xt; v = ^c^-{-2gs; t; = n/'7T644x^ ; 
s = ct-i-ff^ = ot + 16-1 xf; s = ^ ^ = ^015528 {v^ - c% 

If a body be projected vertically to height, with a velocity which 

we shall term o, then the formulas become, 

f 

V = c — 32-2 X ^; v = ^c" — 64-4 xs; s =^ ct — g ^ = 

ct - 16^1 xt^; s = ^^-^-^ = -015528 {c^ - v^). 

17. What space is described by a body passing from 18 feet velo- 
city to 30 feet velocity during its free descent in vacuo. 

From the annexed table, we find that the height due to 30 feet 

velocity = 13-97516 

The height due to 18 = 5-03106 

Space described 8-94410 

Since this problem and table are often required in practical me- 
chanics, we shall enter into more particulars respecting it. 



As s = 



V 



v" 



2a 2a 2 



9 ^9 ^9 

if we put h = height due to the initial velocity c; that is, 

h = n-'f and A^ = the height due to the terminal velocity v ; that is, 

^1 = 2^; then, 

s = h^ — h, for falling bodies, as in the last example ; and 

s = h — h^, for ascending bodies. 
Although these formulas are only strictly true for a free descent 
in vacuo, they may be used in air, when the velocity is not great. 
The table will be found useful in hydraulics, and for other heights 
and velocities besides those set down, for by inspection it is seen 
that the height -201242 answers to the velocity 3*6 ; and the height 
20-12423 to 36 ; and the height 2012-423 to 360 ; and so on. 



WEIGHT — GRAVITY — MASS. 



389 



Table of the Heights corresponding to different Velocities^ in feet 

the second. 



>5 




CORKESPONDIXG HEIGHT IN 


Feet. 


1 





1 


2 


3 


4 


5 


1 ' 


7 


8 1 9 





000000 


•000155 


•000621 


•001398 


•002484 


•003882 


•005590 


•007609 


•009938' •012578 


1 


015528 


•018789 


•020652 


•026242 


•0304348 


•0349379 


•039752 


•044876 


•050311! -056056 


2 


062112 


•0G8478 


•075155 


•082143 -089441 


•097050 


•104969 


•113199] ^121739 -1305901 


3 


139752 


•149224 


•159006 


•169099; -187888 


•190217 


•201242 


•212577 -224224! •236180 


4 


248447 


•261025 


•273913 


•285714 -300621 


•314441 


1 -328572 


-343013! -357764' -372826! 


5 


388199 


•403882 


•419877 


•436180' ^452795 


•469720 


I -486956 


-504503 


•522360 -550578 


6 


559006 


•577795 


•596S94 


•616304 -636025 


•6560G0 


•676397 


-697050 


-718013 -739286 


7 


760870 


•782764 


•804970 


•827484 -850310 


•873447 


1 -896895 


-920652 


-944721 -969099 


8 


993789 


1-018790 


1-044100 


1-069720 1-095652 


1-121895 


1-148421 


1-175311 


1-201482 1-229971 


9 1 


257764 


1-285869 


1-314285 


1-343012 ' 1-372050- 


1-401400 


1-431055 


1-461025! 1-491304' 1-521894| 



The following extension is obtained from the foregoing table, 
by mere inspection, and moving the decimal point as before di- 
rected. 



-&-S 




■fe^ 




^ 




>»*5 




o o 


Corresponding 


"5 « 


Corresponding 


ll 


Corresponding 


"1 


Corresponding 


•1^ 


Height iu Feet. 




Height in Feet. 


oPh 


Height in Feet. 




Height in Feet. 


>.s 




>a 




^.S 




>a 




10 


1-552795 


19 


5-60-559 


28 


12-17392 


37 


21-25777 


11 


1-878882 


20 


6-21118 


29 


13-05901 


38 


22-42236 


12 


2-065218 


21 


6-84783 


30 


13-97516 


39 


23-61802 


13 


2-624224 


22 


^ 7-51553 


31 


14-92237 


40 


24-84472 


14 


3-043478 


23 


8-21429 


32 


15-90062 


41 


26-10249 


15 


3-49379 


24 


8-94410 


33 


16-90994 


42 


27-39131 


16 


3-97516 


25 


9-70497 


34 


18-78883 


43 


28-57143 


17 


4-48758 


26 


10-49690 


35 


19-02174 


44 


30-06212 


18 


5-03106 


27 


11-31988 


36 


20-12423 


45 


31-4441 



18. What mass does a body weighing 30268 lbs. contain ? 

30268 302680 ^_ , 

= 940 lbs. 



32-2 



322 



For the mass is equal to the weight divided by g. And g is 
taken equal to 32*2 ; but the acceleration of gravity is somewhat 
variable ; it becomes greater the nearer we approach the poles of 
the earth. It is greatest at the poles and least at the equator, 
and also diminishes the more a body is above or below the level of 
the sea. The mass, so long as nothing is added to or taken from 
it, is invariable, whether at the centre of the earth or at any dis- 
tance from it. If M be the mass and W the weight of a body, 

W W 

Then M = — = oo-o = '0310559 W. 
g dZ'Z 

19. What is the mass of a body whose weight is 200 lbs ? 
•031055 X 200 = 6-21118 lbs. 

The weight of a body whose mass is 200 lbs. is 32-2 X 200 = 
6440*0 lbs. It may be remarked, that one and the same steel 
spring is differently bent by one and the same weight at different 
places. 

The force which accelerates the motion of a heavy body on an 
inclined plane, is to the force of gravity as the sine of the inclina- 



390 THE PRACTICAL MODEL CALCULATOR. 

tion of the plane to the radius, or as the height of the plane to its 
length. 

The velocity acquired by a body in falling from rest through a 
given height, is the same, whether it fall freely, or descend on a 
plane at whatever inclination. 

The space through which a body will descend on an inclined 
plane, is to the space through which it would fall freely in the same 
time, as the sine of the inclination of the plane to the radius- 

The velocities which bodies acquire by descending along chords 
of the same circle, are as the lengths of those chords. 

If the body descend in a curve, it suffers no loss of velocity. 

The centre of gravity of a hody is a point about which all its 
parts are in equilihrio. 

Hence, if a body be suspended or supported by this point, the 
body will rest in any position into which it is put. We may, there- 
fore, consider the whole weight of a body as centred in this point. 

The common centre of gravity of two or more bodies, is the point 
about which they would equiponderate or rest in any position. If 
the centres of gravity of two bodies be connected by a right line, 
the distances from the common centre of gravity are reciprocally 
as the weights of the bodies. 

If a line be drawn from the centre of gravity of a body, perpen- 
dicular to the horizon, it is called the line of direction, being the 
line that the centre of gravity would describe if the body fell freely. 

The centre of gyration is that part of a hody revolving about an 
axis, into which if the zvhole quantity of matter ivere collected, the 
same moving force would generate the same angular velocity. 

To find the centre of G-yration. — Multiply the weight of the 
several particles by the squares of their distances from the centre 
of motion, and divide the sum of the products by the weight of the 
whole mass ; the square root of the quotient will be the distance 
of the centre of gyration from the centre of motion. 

The distances of the centre of gyration from the centre of mo- 
tion, in different revolving bodies, are as follow : — 

In a straight rod revolving about one end, the length X 'STTS. 

In a circular plate, revolving on its centre, the radius X •7071. 

In a circular plate, revolving about one diameter, the radius X '5. 

In a thin circular ring, revolving about one diameter, radius X 
•7071. 

In a solid sphere, revolving about one diameter, the radius X 
•6325. 

In a thin hollow sphere, revolving about one diameter, radius X 
•8164. 

In a cone, revoh^ng about its axis, the radius of the base X 
•5477. 

In a right-angled cone, revolving about its vertex, the height X 
•866. 



SPECIFIC GRAVITY. 391 

In a paraboloid, revolving about its axis, the radius of the base 
X -5773. 

The centre of 'percussion is that point in a body revolving about 
a fixed axis, into which the whole of the force or motion is collected. 

It is, therefore, that point of a revolving body which would strike 
any obstacle with the greatest effect ; and, from this property, it 
has received the name of the centre of percussion. 

The centres of oscillation and percussion are in the same point. 

If a heavy straight bar, of uniform density, be suspended at one 
extremity, the distance of its centre of percussion is two-thirds of 
its length. 

In a long slender rod of a cylindrical or prismatic shape, the 
centre of percussion is nearly two-thirds of the length from the 
axis of suspension. 

In an isosceles triangle, suspended by its apex, the distance of 
the centre of percussion is three-fourths of its altitude. In a line 
or rod whose density varies as the distance from the point of sus- 
pension, also in a fly-wheel, and in wheels in general, the centre 
of percussion is distant from the centre of suspension three-fourths 
of the length. 

In a very slender cone or pyramid, vibrating about its apex, the 
distance of its centre of percussion is nearly four-fifths of its length. 

Pendulums of the same length vibrate slower, the nearer they 
are brought to the equator. A pendulum, therefore, to vibrate 
seconds at the equator, must be somewhat shorter than at the poles. 

When we consider a simple pendulum as a ball, which is sus- 
pended by a rod or line, supposed to be inflexible, and without 
weight, we suppose the whole weight to be collected in the centre 
of gravity of the ball. But when a pendulum consists of a ball, 
or any other figure, suspended by a metallic or wooden rod, the 
length of the pendulum is the distance from the point of suspension 
to a point in the pendulum, called the centre of oscillation, which 
does not exactly coincide with the centre of gravity of the ball. 

If a rod of iron were suspended, and made to vibrate, that point 
in which all its force would be collected is called its centre of oscil- 
lation, and is situated at two-thirds the length of the rod from the 
point of suspension. 

SPECIFIC GRAVITY. 

The comparative density of various substances, expressed by the 
term specific gravity, affords the means of readily determining the 
bulk from the known weight, or the weight from the known bulk ; 
and this will be found more especially useful, in cases where the 
substance is too large to admit of being weighed, or too irregular 
in shape to allow of correct measurement. The standard with 
which all solids and liquids are thus compared, is that of distilled 
water, one cubic foot of which weighs 1000 ounces avoirdupois ; 



392 THE PRACTICAL MODEL CALCULATOR. 

and the specific gravity of a solid body is determined by the dif- 
ference between its weight in the air, and in water. Thus, 

If the body be heavier than water, it will displace a quantity of 
fluid equal to it in hulk, and will lose as much weight on immersion 
as that of an equal bulk of the fluid. Let it be weighed first, 
therefore, in the air, and then in water, and its weight in the air 
be divided by the difi'erence between the two weights, and the quo- 
tient will be its specific gravity, that of water being unity. 

A piece of copper ore weighs 56| ounces in the air, and 43f 
ounces in water ; required its specific gravity. 
56-25 — 43*75 = 12*5 and 56-25 -j- 12-5 = 4-5, the specific gravity. 

If the body be lighter than water, it will float, and displace a 
quantity of fluid equal to it in 'weight, the bulk of which will be equal 
to that only of the part immersed. A heavier substance must, 
therefore, be attached to it, so that the two may sink in the fluid. 
Then, the weight of the lighter substance in the air, must be added 
to that of the heavier substance in water, and the weight of both 
united, in water, be subtracted from the sum ; the weight of the 
lighter body in the air must then be divided by the difi'erence, and the 
quotient will be the specific gravity of the lighter substance required. 

A piece of fir weighs 40 ounces in the air, and, being immersed 
in water attached to a piece of iron weighing 30 ounces, the two 
together are found to weigh 3*3 ounces in water, and the iron alone, 
25*8 ounces in the water ; required the specific gravity of the wood. 

40 + 25-8 = 65-8 - 3-3 = 62-5; and 40 -^ 62-5 = 0-64, the 
specific gravity of the fir. 

The specific gravity of 2b fluid may be determined by taking a 
solid body, heavy enough to sink in the fluid, and of known spe- 
cific gravity, and weighing it both in the air and in the fluid. The 
difierence between the two weights must be multiplied by the spe- 
cific gravity of the solid body, and the product divided by the 
weight of the solid in the air : the quotient will be the specific 
gravity of the fiuid, that of water being unity. 

Required the specific gravity of a given mixture of muriatic acid 
and water ; a piece of glass, the specific gravity of which is 3, 
weighing 3|: ounces when immersed in it, and 6 ounces in the air. 

6 - 3-75 = 2-25 x 3 = 6-75 -r- 6 = 1-125, the specific gravity. 

Since the weight of a cubic foot of distilled water, at the tem- 
perature of 60 degrees, (Fahrenheit,) has been ascertained to be 
1000 avoirdupois ounces, it follows that the specific gravities of all 
bodies compared with it, may be made to express the weight, in 
ounces, of a cubic foot of each, by multiplying these specific gra- 
vities (compared with that of water as unity) by 1000. Thus, that 
of water being 1, and that of silver, as compared with it, being 
10-474, the multiplication of each by 1000 will give 1000 ounces 
for the cubic foot of water, and 10474 ounces for the cubic foot 
of silver. 



SPECIFIC GRAVITY. 

In the following tables of s^^ecific gravities, the numbers in the 
first column, if taken as whole numbers, represent the weight of a 
cubic foot in ounces ; but if the last three figures are taken as deci- 
mals, they indicate the specific gravity of the body, water being 
considered as unity, or 1. 

To ascertain the number of cubic feet in a substance, from its 
weight, the whole weight in 2^ounds avoirdupois must be divided by 
the figures against the name, in the second column of the table, 
taken as whole numbers and decimals, and the quotient will be the 
contents in cubic feet. 

Required the cubic content of a mass of cast-iron, weighing 7 cwt. 
1 qr. = 812 lbs. 

812 lbs. -^ 450-5 (the tabular weight) = 1-803 cubic feet. 

To find the weight from the measurement or cubic content of a 
substance, this operation must be reversed, and the number o? cubic 
feet, found by the rules given under ^^Mensuration of Solids," 
multiplied by the figures in the second column, to obtain the weight 
in pounds avoirdupois. 

Required the weight of a log of oak, 3 feet by 2 feet 6 inches, 
and 9 feet long. 

9x3x2-5 = 67-5 cubic feet. 

And 67-5 x 58-2 (the tabular weight) = 3928-5 lbs., or 35 cwt. 
qr. 8J lbs. 

The velocity g, which is the measure of the force of gravity, 
varies with the latitude of the place, and with its altitude above 
the level of the sea. 

The force of gravity at the latitude of 45° = 32-1803 feet ; at 

any other latitude L, g = 32-1803 feet - 0-0821 cos. 2 L. If 

g^ represents the force of gravity at the height h above the sea, 

and r the radius of the earth, the force of gravity at the level of the 

5^ 
sea will be ^=/ (1 + ^). 

In the latitude of London, at the level of the sea,^ = 32-191 feet. 
Do. Washington, do. do., g = 32-155 feet. 

The length of a pendulum vibrating seconds is in a constant 
ratio to the force of gravity. 

I = 9-8696044. 

Length of a pendulum vibrating seconds at the level of the sea, in 
various latitudes. 

At the Equator 39-0152 inches. 

Washington, lat. 38° 53' 23'' 39-0958 — 

New York, lat. 40° 42' 40" 39-1017 — 

London, lat. 51° 31' 39-1393 — 

lat. 45° 39-1270 — 

lat.X 39-1270 in.— 0-09982 cos. 2 Z. 



394 



THE PRACTICAL MODEL CALCULATOR. 



Specific Grravity of various Substances. 





Weight of a 


Weight ofa 




Weight of a 


Weight of 3 1 




cubic foot 


cubic foot 




cubic foot 


cubic foot 


METALS. 

Antimony, fused . 


in ounces. 


in pounds. 


STONES. — Continued. 


in ounces. 


in pounds. 


6,624 


414-0 


Grindstone 




2,143 


1340 


Bismuth, cast . 


9,823 


6140 


Gypsum, opaque . 




2,168 


13.5-5 


Brass, common, cast . 


7,824 


4890 


semi-transparent . 








2,306 


144-1 


cast .... 


8,396 


524-8 


Jet, bituminous . 








1,259 


78-3 


wire-drawn 


8,5« 


5340 


Lime-stone 








3.182 


199-0 


Copper, cast 


8,788 


549-2 


Marble . 








2,700 


168-8 


wire-drawn 


8,878 


554-9 


Mill-stone 








2,484 


155-2 


Gold, pure, cast 


19,258 


1203-6 


Porcelain, China . 








2,335 


1491 


22 carats, stand 


17,436 


1093-0 


Portland-stoue 








2,570 


160-6 


20 carats, trinket . 


15,709 


982-0 


Pumice stone 








915 


57-2 


Iron, cast 


7,207 


450-5 


Paving-stone . 








2,416 


151-0 


bars .... 


7,78.S 


486-8 


Purbeck-stone 








2,601 


162-6 


Lead, cast . 


11,352 


709-5 


Rotten-stone 








1,981 


124-0 


litharge .... 


6,300 


393-8 


Slate, common 








2,672 


167-0 


Manganese . 


7,000 


437-5 


new 








2,854 


178-4 


Mercury, solid,-) 
40O below OO ; • • 


15,632 


977-0 


Stone, common . 
rag 








2,520 
2,470 


157-5 
154-4 


at 32 deg. Fahr. 


J^'^P 


851-2 


Sulphur, native . 








2,033 


127-1 


ateOdeg. 


}?'^^ 


848-8 


melted . 




1,991 


124-5 


at 212 deg. 


13,375 


836-8 










Nickel, cast 

Platina, crude, grains . 


7,807 
15.602 


4880 
975-1 


LIQUIDS. 








purified .... 


19,500 


1218-8 


'Acetic acid 




1,007 


63-0 


hammered 


20,337 


1271-1 


Acetous acid 




1,025 


641 


rolled .... 


22,069 


1379-4 


Alcohol, commercial 








837 


52-3 


wire-drawn 


21,042 


1315-1 


highly rectified . 








829 


51-8 


Silver, cast, pure . 


10,474 


654-6 


Ammonia, liquid 
Beer 






_ 


897 


56-1 


Parisian standard . 


10,175 


6315-0 








1,023 


680 


French coin . 


10,048 


6280 


Ether, sulphuric 
Milk of cows 








739 


46-2 


shilling, Geo. III. . 


^^Ati 


658-4 










64-5 


Steel, soft .... 


J'E 


489-6 


2*Iuriatic acid . 








i;i94 


74-6 


hardened . 


7'§^ 


490-0 


Nitric acid 








1,271 


79-5 


tempered 


7,816 


438-5 


highly concentrated 








1,533 


99-0 


tempered and h 


ard • 


7,818 


48S-6 


Oil of almonds, sweet 








917 


57-4 


Tin, pure Cornish 




7,291 


455-6 


hemp-seed . 
linseed 








926 


58-0 


Tungsten . 




^•*J'J^ 


379-1 








940 


58-8 


Uranium . 




6,440 


402-5 


olives 








915 


57-3 


Wolfram 




7,119 


445-0 


poppies 
rape-seed 








924 


57-8 


Zinc, usual state 


. 


^'?^? 


429-0 








919 


57-5 


pure . . 




7,191 


449-5 


turpentine, essence 
whales . 








870 
923 


54-4 

57-3 


•WOODS 








Spirits of wine, "1 
commercial/' 








837 


52-4 


Ash- . 




845 


52-9 


highly rectified . 








829 


51-9 


Beech . 




852 


53-2 


Sulphuric acid 








1,841 


1151 


Box, Dutch 




912 


57-0 


highly concentrated 




r 




2,1-25 


1330 


French 




1,323 


83-0 


Turpentine, liquid 








991 


62-0 


Brazilian 




1,031 


64-5 


Vinegiir, distilled . 


. 


1,010 


631 


Cedar, American 




561 


&5-1 


Water, rain, or distilled 




1,000 


62-5 


Indian . 




1,315 


82-2 


sea .... 




1,026 


64-1 


Cherry-tree . 




715 


44-8 








Cocoa 




1,040 


650 


MISCELLANEOUS SUB- 






Cork 




240 


150 








Ebony, Indian . 




1,209 


75-6 


STANCES. 






American 
Elm . 
Fir, yellow 

white 
Lignum-vitse 




1,331 
671 
657 
569 

1,333 


83-2 
420 
41-1 
.35-6 
83-4 


Beeswax 

Butter 

Camphor 

Fat, beef or mutton . 

hogs' 

Honey 

Indigo 

Ivory 

Lard 

Opium 

Spermaceti .... 
Sugar, white. . . . 
Tallow 


965 
942 
989 
923 
937 


60-4 
59-0 
620 
57-8 
58-6 


Lime-tree . 
Logwood 
Mahogany 
Maple 




604 

913 

1,063 

750 


37-8 
57-1 

47-0 


1,450 
769 

1,826 
948 


90-6 
43-1 
114-1 
59-2 


Oak, heart of, old 

dry . 
Vine . 
Walnut 
Willow . . 




1,170 
932 

1.327 
671 
535 


73-1 
58-2 
830 
42-0 
36-6 


1,336 
943 

1,606 
942 


83-5 ■ 
59-0 
10O-4 
59-0 


Yew . . 




807 


50-5 












GASES. 




STONES, EARTHS, ETC. 






Atmospheric air being estimated 




Alabaster, yellow 


2,699 


168-8 


asl. 




* white .... 


2;730 


170-6 






Borax .... 


1,714 


107-1 


Atmospheric, or common air . . . 


1-000 


Brick earth 




2,000 


125-0 


Ammoniacal gas 


•590 


Chalk . 




2,784 


174-0 


Azote 


-969 


Coal, Cannel . 




1,270 


79-4 


Carbonic acid 


1.520 


Newcastle 




1,270 


79-4 


Carbonic oxide 


•960 


Staffordshire 




1,240 


77-5 


Cavburetted hydrogen .... 


•491 


Scotch 




1,300 


81-2 


Chlorine 


•470 


Emery . . 




4,000 


2500 


Hydrogen 


•074 


Flint, black . 




2,582 


1620 


Muriatic acid gas 


1-278 


Glass, flint 




2,933 


170-9 


Nitrous gas 


1-094 


white 




2,892 


168-2 


Nitrous acid gas 


2-427 


Granite, Aberd. b 


lue * . 


2.625 


164-1 


Oxvgen 


1-104 


Cornish . 




2,662 


166-4 


Steam 


-690 


Egyptian, red 


2,654 


165-9 


Sulphuretted hydrogen .... 


1-777 


gray 


• 


2,728 


170-5 


Sulphurous acid . 








. 


2193 



SPECIFIC GRAVITY. 



395 



Table of the Weight of a 


Foot in 


length of 


Flat and Rolled Iron. 


fit 






BREADTH IN INCHES 


iND PARTS OF AN INCH. 




4 


3f 


H 


H 


3 


H 


2^ 


2i 


2 ,lf 


li 


1| 


u 


1 


f 


i 


i 


1-68 


1-57 


1-47 


1-36 


1-26 


1-15 


1-05 


0-94 


0-84 


0-73 


0-63 


0-57 


0-52 


0-42 


0-31 


0-21 


A 


2-52 


2-36 


2-20 


2-04 


1-89 


1-73 


1-57 


1-41 


1-26 


1-10 


0-94 


0-86 


0-78 


0-63 


0-47 


0-31 


1 


3-36 


3-16 


2-94 


2-73 


2-52 


2-31 


2-10 


1-89 


1-68 


1-47 


1-26 


1-18 


1-05 


0-84 


0-63 


0-42 


* 


5-04 


4-72 


4-41 


4-09 


3-78 


3-46 


3-15 


2-83 


2-52 


2-20 


1-89 


1-73 


1-57 


1-26 


0-94 


0-63 


i 


6-72 


6-30 


5-88 


5-46 


604 


4-62 


4-20 


3-78 


3-36 


2-94 


2-52 


•2-31 


2-10 


1-68 


1-26 




t 


8-40 


7-87 


7-35 


6-82 


6-30 


5-77 


5-25 


4-72 


4-20 


3-67 


3-15 


-2-88 


2-62 


2-10 


1-57 




* 


10-08 


9-45 


8-82 


8-19 


7-56 


6-93 


6-30 


6-66 


5-04 


4-41 


3-78 


3-46 


3-15 


2-52 








11-76 


11-02 


10-29 


9-45 


8-82 


8-08 


7-35 


6-61 


5-88 


5-14 


4-41 


4-04 


3-67 


'2-94 






1 


13-44 


12-60 


11-76 


10-92 


10-08 


9-24 


8-40 


7-56 


6-72 


5-87 


5-04 


4-6?, 


4-20 








H 


15-12 


14-16 


13-20 


12-28 


11-34 


10-39 


9-45 


8-50 


7-56 


6-60 


5-67 


5-19 


4-72 








If 


16-80 


15-75 


14-70 


13-65 


12-60 


11-55 


10-50 


9-45 


8-40 


7-35 


6-30 


5-77 










i| 


18-48 


17-32 


16-16 


15-01 


13-86 


12-70 


11-55 


10-39 


9-24 


8-07 














H 


20-18 


18-90 


17-64 


16-38 


15-12 


13-86 


12-CO 


11-34 


10-08 


S-80 














U 


•23-54 


22-05 


20-58 


19-11 


17-64 


16-17 


14-70 


13-^2 


















2 


26-88 


25-20 


23-52 


21-84 


20-16 


18-48 


16-80 


15-12 


















2^ 


33-65 


31-50 


29-40 


27-39 


25-20 


23-10 






















3 


40-32 


37-80 


35-28 


32-76 


























3i 


47-04 



































Table of the Weight of Cast-iron Pipes, 


in length 


S. 


i 


M 


§ 


Weight. 


i 


1 


§ 


Weight. 


i 


^ 


g 


Weight. 


« 


^ 


*^ 




M 


H 


1-1 




n 


^ 


>3 




lach. 


Inch. 


Feet. 


C. qr. lb. 


Inch. 


Inch. 


Feet. 


C. qr. lb. 


Inch. 


Inch. 


Feet. 


C. qr. lb. 


1 


i 


3i 


12 


6i 


§ 


9 


2 16 


Hi 




9 


5 7- 




f 


3^ 


21 




i 


9 


2 3 20 






9 


6 1 12 


U 


i 


4i 


21 




§ 


9 


3 2 21 






9 


■ 7 2 8 




i 


4i 


1 4 






9 


4 1 21 






9 


10 1 2 


2 


i 


6 


1 8 






9 


6 14 


12 




9 


5 24 




S 


6 


2 


7 




9 


3 7 






9 


6 2 8 


2i 


i 


6 


1 16 






9 


3 3 20 






9 


7 3 20 




f 


6 


2 10 






9 


4 3 5 






9 


10 3 




i 


6 


3 10 






9 


6 2 4 


12i 




9 


5 1 16 


3 


I 


9 


2 20 


7i 




9 


3 16 






9 


6 3 9 




f 


9 


1 6 






9 


4 22 






9 


8 10 




i 


9 


1 1 12 






9 


5 10 






9 


11 21 




f 


9 


13 6 






9 


7 


13 




9 


5 2 20 




I 


9 


2 10 


8 




9 


3 2 4 




? 


9 


7 14 


3* 


I 


9 


3 






9 


4 1 25 






9 


8 2 7 




t 


9 


1 21 






9 


5 1 18 




1 


9 


11 2 12 




i 


9 


1 2 14 






9 


7 1 16 


13i 


, 


9 


5 3 7 




§ 


9 


2 8 


8i 




9 


3 3 2 






9 


7 1 12 




1 


9 


2 2 






9 


4 2 26 






9 


8 3 16 


4 


i 


9 


1 1 10 






9 


5 2 22 






9 


11 3 24 




i 


9 


1 3 12 






9 


7 3 8 


14 




9 


6 4 




i 


9 


2 1 12 


9 




9 


4 






9 


7 2 16 




I 


9 


2 2 21 






9 


5 4 






9 


9 10 


4i 


t 


9 


12 2 






9 


6 2 






9 


12 1 14 




i 


9 


2 4 






9 


8 26 


14i 




9 


6 24 




t 


9 


2 2 14 


n 




9 


4 18 






9 


7 3 14 




i 


9 


3 21 






9 


5 1 






9 


9 2 2 


5 


t 


9 


1 2 22 






9 


6 1 6 






9 


12 3 6 




i 


9 


2 1 10 






9 


8 2 20 


15 




9 


6 1 21 




t 


9 


2 3 17 


10 




9 


4 1 10 






9 


9 3 7 




1 


9 


3 1 24 






9 


5 1 26 






9 


13 26 


5i 


f 


9 


1 3 10 






9 


4 2 14 




li 


9 


16 3 5 




i 


9 


2 2 






9 


9 8 


15i 




9 


6 2 14 




# 


9 


3 18 


lOi 




9 


4 2 14 






9 


10 9 10 




1 


9 


3 3 7 






9 


5 3 7 






9 


13 2 17 




1 


9 


5 12 






9 


7 




n 


9 


17 1 6 


6 


I 


9 


2 






9 


9 2 


16 




9 


7 22 




i 


9 


2 2 21 


11 




9 


4 3 14 






9 


10 1 20 




f 


9 


3 1 17 






9 


6 11 






9 


14 8 




I 


9 


4 16 






9 


7 1 7 




li 


9 


17 3 14 




1 


9 


5 2 20 






9 


9 3 20 




li 


9 


21 3 4 



396 



THE PRACTICAL MODEL CALCULATOR. 



Table of the Weight of one 


Foot Lengt 


h of 3faUeahle Iron. 


SQTJ-iEE IBOX. 




KOCXl 


3 IROX. 


ScantUng. 


Weight. 


Diameter. 


Weight. 


Circumference. Weight. 


Inches. 


Potinds. 


Inches. 


Pounds. 


Inches. | Pounds. 


1 


0-21 


i 


0-16 


1 ' 0-26 


3 


0-47 


•| 


0-37 


li 0-41 


1 


0-84 


^ 


0-66 


IJ 


0-59 


|. 


1-34 


1^ 


1-03 


If 


0-82 


a 


1-89 


1 


1-48 


2 


1-05 


i 


2-57 


¥ 


2-02 


21 


1-34 




3-36 




2-63 


^ 


1-65 


H 


4-25 


H 


3-33 


23. 


2 01 


l| 


5-25 




4-12 


3 


2-37 


If 


6-35 


1| 


4-98 


H 


2-79 


- ^ 


7-56 


1^ 


5-93 


H 


3-24 




8-87 


lA 


6-96 


3f 


3-69 


If 


10-29 


1|. 


8-08 


4 


4-23 


17 


11-81 


1|. 


9-27 


^ 


5-35 


2 


13-44 


2 


10-55 


5 


6-61 


2i 


17-01 


2^ 


13-35 


5J 


7-99 


^ 


21-00 


2i 


16-48 


6 


9-51 


2f 


25-41 


2f 


19-95 


6^ 


11-18 


3 


30-24 


3 


23-73 


7 


12-96 


H 


41-16 


3-1- 


27-85 


7i 


14-78 


4 


53-76 


3^ 


32-32 


8 


16-92 


^ 


68-04 


3| 


37-09 


H 


19-21 


5 


84-00 


4 


42-21 


9 


21-53 


6 


120-96 


4i 


53-41 


10 


26-43 


7 


164-64 


5 


65-93 


i 12 


31-99 



The following tables are rendered of great utility by means of 
this table : — 



The weight of Water 
Copper 



being 



Brass 

Iron, cast = 

Lead = 

Zinc = 

Gun-metal = 

Sand = 

Coal == 

Brick == 

Stone = 
Timber, average = 



1- 
8-8 
8-4 
7-2 
11-3 
7-2 
8-7 
1-5 
1-25 
2-0 
2-5 
0-85 



Suppose it be required to ascertain the weight of a cast iron 
pipe 26J inches outside and 23| inside, the length being 6} feet. 
Opposite 26J in the table is 

234-85T6 x T-2 x 6-5 = 10991-135. 
And opposite 23| in the table is 

192-2856 X 7-2 x 6-5 = 8998-966 subtract 

1992-169 lbs. avr. 
The succeeding table contains the surface and solidity of spheres, 
together with the edge or dimensions of equal cubes, the length 
of equal cylinders, and the weight of water in avoirdupois pounds : — 



SPECIFIC GRAVITY. 



o97 



Surface and Solidity of Spheres. 



Diameter. 


Surface. 


Solidity. 


Cube. 


CyUnder. 


Water in lbs. 


lin. 


3-1416 


-5236 


•8060 


-6666 


-0190 


tV 


3-5465 


•6280 


•8563 


•7082 


•0227 


i 


3-9760 


-7455 


-9067 


•7500 


-0270 


1% 


4-4301 


•8767 


•9571 


•7917 


-0317 


i 


4-9087 


1-0226 


1-0075 


•8333 


-0370 


A 


5-4117 


1-1838 


1-0578 


•8750 


-0428 


1 


5-9395 


1-3611 


1-1082 


•9166 


-0500 


i\ 


6-4918 


1-5553 


1-1586 


•9583 


-0563 


i 


7-0686 


1-7671 


1-2090 


1-0000 


-0640 


A 


7-6699 


2-0000 


1-2593 


1-0416 


-0723 


1 


8-2957 


2-2467 


1-3097 


1-0833 


-0813 


H 


8-9461 


2-5161 


1-3601 


1-1349 


•0910 


f 


9-6211 


2-8061 


1-4105 


1-1666 


-1015 


if 


10-3206 


3-1176 


1-4608 


1-2083 


•1128 


i 


11-0446 


3-4514 


1-5112 


1-2500 


-1250 


if 


11-7932 


3-8081" 


1-5616 


1-2916 


-1377 


2 in. 


12-5664 


4-1888 


1-6020 


1-3333 


-1516 


tV 


13-3640 


4-5938 


1-6633 


1-3750 


-1662 


i 


14-1862 


5-0243 


1-7127 


1-4166 


-1818 


A 


15-0330 


5-4807 


1-7631 


1-4582 


•1982 


i 


15-9043 


6-9640 


1-8135 


1-5000 


•2160 


/o 


16-8000 


6-4749 


1-8638 


1-5516 


•2342 


1 


17-7205 


7-0143 


1-9142 


1-5832 


•2540 


tV 


18-6655 


7-5828 


1-9646 


1-6250 


•2743 


i 


19-6350 


8-1812 


2-0150 


1-6666 


-2960 


A 


20-6290 


8-8103 


2-0653 


1-7082 


-3187 


1 


21-6475 


9-4708 


2-1157 


1-7500 


-3426 


ii 


22-6907 


10-1634 


2-1661 


1-7915 


•3676 


f 


23-7583 


10-8892 


2-2165 


1-8332 


-3939 


if 


24-8505 


11-6485 


2-2668 


1-8750 


•4213 


i 


25-9672 


12-4426 


2-3172 


1-9165 


•4501 


if 


27-1084 


13-2718 


2-3676 


1-9582 


•4800 


3 in. 


28-2744 


14-1372 


2-4180 


2-0000 


•5114 


1^8- 


29-4647 


15-0392 


2-4683 


2-0415 


•5440 


i 


30-6796 


15-9790 


2-5187 


2-0832 


-5780 


l\ 


31-9191 


16-9570 


2-5691 


2-1250 


-6133 


\ 


33-1831 


17-9742 


2-6195 


2-1665 


-6401 


A 


35-3715 


19-0311 


2-6698 


2-2082 


-6884 


i 


35-7847 


20-1289 


2-7202 


2-2500 


•7281 


tV 


37-1224 


21-2680 


2-7706 


2-2915 


-7693 


^ 


38-4846 


22-4493 


2-8210 


2-3332 


-8120 


A 


39-8713 


23-6735 


2-8713 


2-3750 


-8561 


1 


41-2825 


24-9415 


2-9217 


2-4166 


-9021 


ii 


42-7183 


26-2539 


2-9712 


2-4582 


-9496 


1 


44-1787 


27-6117 


3-0225 


2-5000 


-9987 


if 


45-6636 


29-0102 


3-0728 


2-5415 


1-0493 


i 


47-1730 


30-4659 


3-1232 


2-5832 


1-1020 


if 


48-7070 


31-9640 


3-1730 


2-6250 


1-1561 


4in. 


50-2656 


33-5104 


3-2240 


2-6665 


1-1974 


tV 


51-8486 


35-1058 


3-2743 


2-7082 


1-2698 


i 


53-4562 


36-7511 


3-3247 


2-7500 


1-3293 


A 


55-0884 


38-4471 


3-3751 


2-7915 


1-3906 


i 


56-7451 


40-1944 


3-4255 


2-8332 


1-4538 


tV 


58-4262 


42-0461 


3-4758 


2-8750 


1-5208 


1 


60-1321 


43-8463 


3-5262 


2-9165 


1-5860 


- 


61-8625 


45-7524 


3-5766 


2-9582 


1-6550 



398 



THE PRACTICAL MODEL CALCULATOR. 



Diameter. 


Surface. 


Solidity. 


Cube. 


Cylinder. 


Water in lbs. 


i 


63-6174 


47-7127 


3-6270 


3-0000 


1-7258 


A 


65-3968 


49-7290 


3-6773 


3-0415 


1-7987 , 


1 


67-2007 


51-8006 


3-7277 


3-0832 


1-8736 


¥ 


69-0352 


53-9290 


3-7781 


3-1250 


1-9506 


f 


70-8823 


56-1151 


3-8285 


3-1665 


2-0297 


^^ 


72-7599 


58-3595 


3-8788 


3-2080 


2-1109 


i 


74-6620 


60-6629 


3-9292 


3-2500 


2-1942 


H 


76-5887 


62-9261 


3-9796 


3-2913 


2-2760 


5 m. 


78-5400 


65-4500 


4-0300 


3-3332 


2-3673 


T> 


80-5157 


67-9351 


4-0803 


3-3750 


2-4572 


i 


82-5160 


70-4824 


4-1307 


3-4155 


2-5453 


1% 


84-5409 


73-0926 


4-1811 


3-4582 


2-6438 


i 


86-5903 


75-7664 


4-2315 


3-5000 


2-7605 


A 


88-6641 


78-5077 


4-2818 


3-5414 


2-8396 


1 


90-7627 


81-3083 


4-3322 


3-5832 


2-9407 


tV 


92-8858 


84-1777 


4-3820 


3-6250 


3-0447 


^ 


95-0334 


87-1139 


4-4330 


3-6665 


3-1509 


A 


97-2053 


90-1175 


- 4-4633 


3-7080 


3-2595 


1 


99-4021 


93-1875 


4-5337 


3-7500 


3-3706 


fi 


101-6233 


96-3304 ' 


4-5841 


3-7913 


3-4843 


1 


103-8691 


99-5412 


4-6345 


3-8330 


3-6004 


II 


106-1394 


102-8225 


4-6848 


3-8750 


3-7191 


i 


108-4342 


106-1754 


4-7352 


3-9163 


3-8404 


if 


110-7536 


109-5973 


4-7856 


3-9580 


'3-9641 


6 in. 


113-0976 


113-0976 


4-8360 


4-0000 


4-0907 


T^B- 


115-4660 


116-6688 


4-8863 


4-0417 


4-2200 


i 


117-8590 


120-3139 


4-9367 


4-0833 


4-3517 


1% 


120-2771 


124-0374 


4-9871 


4-1250 


4-4874 


i 


122-7187 


127-8320 


5-0375 


4-1666 


4-6236 


/f 


125-1852 


. 131-7053 


5-0878 


4-2083 


4-7638 


1 


127-6765 


135-6563 


5-1382 


4-2500 


4-9067 


tV 


130-1923 


139-6854 


5-1886 


4-2917 


5-0524 


J 


132-7326 


143-7936 


5-2390 


4-3332 


5-2010 


A 


135-2974 


147-9815 


5-2893 


4-3750 


5-3525 


f 


137-8867 


152-2499 


5-3377 


4-4165 


5-5069 


H 


140-5006 


156-5997 


5-3901 


4-4583 


5-6786 


f 


143-1391 


161-0315 


5-4405 


4-5000 


5-8245 


H- 


145-8021 


167-5461 


5-4908 


4-5416 


6-0601 


i 


148-4896 


170-1682 


5-5412 


4-5832 


6-1550 


if 


151-2017 


174-8270 


5-5916 


4-6250 


6-3235 


7 in. 


153-9384 


179-5948 


5-6420 


4-6665 


6-4960 


tV 


156-6995 


184-4484 


5-6923 


4-7082 


6-6725 


i 


159-4852 


189-3882 


5-7427 


4-7500 


6-8502 


A 


162-2955 


194-1165 


5-7931 


4-7915 


7-0212 


i 


165-1303 


199-5325 


5-8435 


4-8332 


7-2171 


fV 


167-9895 


204-7371 


5-8938 


4-8750 


7-4053 


1 


170-8735 


210-0331 


5-9442 


4-9166 


7-5970 


tV 


173-7520 


215-4172 


5-9946 


4-9582 


7-7916 


# 


176-7150 


220-8937 


6-0450 


5-0000 


7-9897 


t'? 


179-6725 


226-7240 


6-0953 


5-0415 


8-2006 


1 


182-6545 


232-1235 


6-1457 


5-0832 


8-3960 


ii 


185-6611 


237-8883 


6-1961 


5-1250 


8-6044 


1 


188-6923 


243-7276 


6-2465 


5-1665 


8-8157 


if 


191-7480 


249-4720 


6-2968 


5-2082 


9-0234 


¥ 


194-8282 


255-7121 


6-3472 


5-2500 


9-2491 


if 


197-9330 


261-9673 


■ 6-3976 


5-2913 


9-4753 


Sin. 


201-0624 


268-0832 


6-4480 


5-3330 


9-6965 - 


I 


204-2162 


274-4156 


6-4983 


5-3750 


9-9260 



SPECIFIC GRAVITY. 



399 



Diameter. 


1 Surface. 


Solidity. 


Cube. 


Cylinder. 


Water in lbs. 


i 


207-3946 


280-8469 


6-5487 


5-4164 


10-1583 


l\ 


210-5976 


287-3780 


6-5991 


5-4581 


10-3944 


i 


213-8251 


294-0095 


6-6495 


5-5000 


10-6343 


A 


217-0770 


300-7422 


6-6998 


5-5414 


10-8778 


f 


220-3537 


307-5771 


6-7502 


5-5831 


11-1250 


tV 


223-6549 


314-5147 


6-8006 


5-6250 


11-3760 


^ 


226-9806 


321-5553 


6-8510 


5-6664 


11-6306 


1% 


230-3308 


328-7012 


6-9013 


5-7080 


11-8891 


i 


233-7055 


335-9517 


6-9517 


5-7500 


12-1514 


H 


237-1048 


343-3079 


7-0021 


5-7913 


12-4170 


f 


240-5287 


350-7710 


7-0525 


5-8330 


12-6874 


fl 


243-9771 


358-3412 


7-1028 


5-8750 


12-9612 


i 


247-4500 


366-0199 


7-1532 


5-9163 


13-2390 


if 


250-9475 


373-8073 


7-2036 


5-9580 


13-5206 


9 in. 


254-4696 


381-7017 


7-2540 


6-0000 


13-8062 


A 


258-0261 


389-7118 


7-3043 


6-0417 


14-0959 


i 


261-5872 


397-8306 


7-3547 


6-0833 


14-3895 


1^ 


265-1829 


406-0613 


7-4051 


6-1250 


14-6872 


i 


268-8031 


414-4048 


7-4555 


6-1667 


14-9890 


A 


272-4477 


421-2907 


7-5058 


6-2083 


15-2381 


1 


276-1171 


. 431-4361 


7-5562 


6-2500 


15-6050 


tV 


279-8110 


440-1294 


7-6066 


6-2916 


15-9195 


i 


283-5294 


448-9215 


7-6570 


6-3333 


16-2375 


1% 


287-2723 


457-8500 


7-7073 


6-3750 


16-5604 


1 


291-0397 


466-8763 ' 


7-7557 


6-4166 


16-6869 


li 


294-8310 


476-0304 


7-8081 


6-4582 


17-2180 


4 


298-4483 


485-3035 


7-8585 ■ 


6-5000 


17-5534 


if 


302-4894 


494-6952 


7-9088 


6-5415 


17-8931 


i 


306-3550 


504-2094 


7-9592 


6-5832 


18-2373 


if 


310-9452 


513-8436 


8-0096 


6-6250 


18-5857 


10 in. 


314-1600 


523-6000 


8-0600 


6-6666 


18-6786 


tV 


318-0992 


533-4789 


8-1103 


6-7083 


19-2960 


? 


322-0630 


543-4814 


8-1607 


6-7500 


19-6577 


/^ 


326-0514 


553-6081 


8-2111 


6-7916 


20-0240 


i 


330-0643 


563-8603 


8-2615 


6-8333 


20-3948 


A 


334-1016 


574-2371 


8-3118 


6-8750 


20-6682 


1 


338-1637 


584-7415 


8-3622 


6-9166 


21-1501 


iV 


342-2503 


595-3677 


8-4126 


6-9582 


21-5344 


# 


346-3614 


606-1318 


8-4630 


7-0000 


21-9238 


A 


350-4970 


617-0207 


8-5133 


7-0416 


22-3176 


1 


354-6571 


628-0387 


8-5637 


7-0833 


22-7162 


ii 


358-8418 


639-1871 


8-6141 


7-1250 


23-1194 


f 


363-0511 


650-4666 


8-6645 


7-1666 


23-5274 


if 


367-2849 


661-8580 


8-7148 


7-2082 


23-9394 


i 


371-5432 


673-4222 


8-7652 


7-2500 


24-3577 


if 


375-8261 


685-0997 


8-8156 


7-2915 


24-7801 


11 in. 


380-1336 


696-9116 


8-8660 


7-3330 


25-2073 


A 


384-4655 


708-9106 


8-9163 


7-3750 


25-6414 


? 


388-8220 


720-9409 


8-9667 


7-4165 


26-0764 


A 


393-2031 


733-1599 


9-0171 


7-4582 


26-5184 


? 


397-6087 


745-5004 


9-0675 


7-5000 


26-5657 


A 


402-0387 


758-0104 


9-1178 


7-5414 


27-4162 


1 


406-4935 


770-6440 


9-1682 


7-5832 


27-8742 


tV 


410-7728 


783-5787 


9-2186 


7-6250 


28-3420 


/ 


415-4766 


796-3301 


9-2690 


7-6664 


28-8033 


A 


420-0049 


809-3844 


9-3193 


7-7080 


29-2754 


f 


424-5576 


822-5807 


9-3697 


7-7500 


29-7527 


f^ 


429-1351 


835-9695 


9-4201 


7-7913 


30-2370 



400 



THE PRACTICAL MODEL CALCULATOR. 



Diameter. 


Surface. 


Solidity. 


Cube. 


CyUnder. 


AVater in lbs. 


f 


433-7371 


849-4035 


9-4705 


7-8330 


30-7229 


ii- 


438-3636 


863-0283 


9-5208 


7-8750 


31-2157 


i 


443-0146 


876-7999 


9-5772 


7-9163 


31-3883 


if 


447-6902 


890-7070 


9-6216 


7-9580 


32-2169 


12 in. 


452-3904 


904-7808 


9-6720 


8-0000 


32-7259 


i 


471-4363 


962-5158 


9-8735 


8-1666 


34-8142 


^ 


490-8750 


1022-656 


10-0750 


8-3332 


36-9886 


f 


506-7064 


1085-251 


10-2765 


8-5000 


39-2535 


13 in. 


530-9304 


1150-337 


10-4780 


8-6666 


41-6077 


i 


551-5471 


1218-000 


10-6790 


8-8332 


44-0551. 




572-5566 


1288-252 


10-8810 


9-0000 


46-5961 


1 


593-9587 


1361-346 


11-0825 


9-1665 


49-2399 


14 in. 


615-7536 


1436-758 


11-2840 


9-3332 


51-9675 


i 


637-9411 


1515-106 


11-4855 


9-5000 


54-8014 


i 


660-5214 


1596-260 


11-6870 


9-6665 


57-7367 


f 


683-4943 


1680-265 


11-8885 


9-8332 


60-7751 


15 in. 


706-8600 


1767-150 


12-0900 


10-0000 


64-0178 


i 


730-6183 


1856-988 


12-2915 


10-1666 


67-1672 


^ 


754-7694 


1949-821 


12-4930 


10-3332 


70-5250 


f 


779-3131 


2045-697 


12-6940 


10-5000 


73-9929 


16 in. 


804-2496 


2144-665 


12-8960 


10-6666 


77-5725 



Table containing the Weight of Flat Bar Iron^ 1 foot in length, 







of various breadths and thicknesses. 






B . 

11 


THICKNESS IN PAKTS OF AN INCH. 


i 


1% 


1 


tV 


J 


._A_ 


1 


f 


t 


1 inch. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


lin. 


0-83 


1-04 


1-25 


1-45 


1-66 


1-87 


2-08 


2-50 


2-91 


3-33 


H 


0-93 


1-17 


1-40 


1-64 


1-87 


2-00 


2-34 


2-81 


3-28 


8-75 


li 


1-04 


1-30 


1-56 


1-82 


2-08 


2-34 


2-60 


3-12 


3-74 


4-16 


If 


M4 


]-43 


1-71 


2-00 


2-29 


2-57 


2-86 


3-43 


4-01 


4-58 


1| 


1-25 


1-56 


1-87 


2-18 


2-50 


2-81 


3-12 


3-75 


4-37 


5-00 


If 


1-85 


1-69 


2-03 


2-36 


2-70 


3-04 


3-38 


4-06 


4-73 


5-41 


If 


1-45 


1-82 


2-18 


2-55 


2-91 


3-28 


3-64 


4-37 


5-10 


5-83 


1| 


1-56 


1-95 


2-34 


2-73 


3-12 


3-51 


3-90 


4-68 


5-46 


6-25 


2 in. 


1-66 


2-08 


2-50 


2-91 


3-33 


3-75 


4-16 


5-00 


5-83 


6-66 


2i 


1-77 


2-21 


2-65 


3-09 


3-54 


3-98 


4-42 


6-31 


6-19 


7-08 


2| 


1-87 


2-34 


2-81 


3-28 


3-75 


4-21 


4-68 


5-62 


6-56 


7-50 


2t 


1-97 


2-47 


2-96 


3-46 


3-95 


4-45 


4-94 


5-93 


6-92 


7-91 


2| 


2-08 


2-60 


3-12 


3-64 


4-16 


4-68 


5-20 


6-25 


7-29 


8-33 


2% 


2-18 


2-73 


3-28 


3-82 


4-37 


4-92 


5-46 


6-56 


7-65 


8-75 


2^ 


2-29 


2-86 


3-43 


401 


4-58 


5-15 


5-72 


6-87 


8-02 


9-16 


2|. 


2-39 


2-99 


3-59 


4-19 


4-79 


5-39 


5-98 


7-18 


8-38 


9-58 


Sin. 


2-50 


3-12 


3-7o 


4-37 


5-00 


5-62 


6-25 


7-50 


8-75 


10-00 


3-1- 


2-70 


3-38 


4-06 


4-73 


5-41 


6-09 


6-77 


8-12 


9-47 


10-83 


^t 


2-91 


3-64 


4-37 


5-10 


5-83 


6-56 


7-29 


8-75 


10-20 


11-66 




3-12 


3-90 


4-68 


5-46 


6-25 


7-03 


7-81 


9-37 


10-93 


12-50 


4'in. 


3-33 


4-16 


5-00 


5-83 


6-66 


7-50 


8-33 


10-00 


11-66 


13-33 


4i 


3-54 


4-42 


5-31 


6-19 


7-08 


7-96 


8-85 


10-62 


12-39 


14-16 


4i. 


3-75 


4-68 


5-62 


6-56 


7-50 


8-43 


9-37 


11-25 


13-12 


15-00 


4f 


3-95 


4-94 


5-93 


6-92 


7-91 


8-90 


9-89 


11-87 


13-85 


15-33 


5 in. 


4-17 


5-20 


6-25 


7-29 


8-33 


9-37 


10-41 


12-50 


14-58 


16-66 


b\ 


4-37 


5-46 


6-56 


7-65 


8-75 


9-84 


10-93 


13-12 


15-31 


17-50 


^ 


4-58 


5-72 


6-87 


8-02 


9-16 


10-31 


11-45 


13-75 


16-04 


18-33 


5| 


4-79 


5-98 


7-18 


8-38 


9-58 


10-78 


11-97 


14-37 


16-77 


19-16 


6 in. 


5-00 


6-26 


7-50 


8-75 


10-00 


11-25 


12-50 


15-00 


17-60 


20-00 



SPECIFIC GRAVITY. 



401 



Table combining tlie Speclfie Gravities and other Properties of 
Bodies. Water the standard of comparison^ or 1000. 



Names. 




METALS. j 


Names. 


STOKES, EAKTHS, ETC.! 


■S 111 i 


ill 

lit 


1 

k 

Si 


1 
1. 


1 


1 

li- 
11 

a 




! 
i 


1 

1 


1 

a 
I 


4 
1.1 

H 


Platinum . . 
Pure Gold . 
Mercury . . 
Lead.. . . 
Pare Silver . 
Bismuth . . 
Copper, cast . 

" wrought 
Brass, cast . 

" sheet . 
Iron, cast . 

" bar . . 
Steel, soft . 

" hard . 

Tin, cast . . 
Zinc, cast . . 


19500 
19258 
13500 
11352 
10474 
8923 
878S 
8910 
7824 
8396 
7264 
7700 
7833 
7816 
7291 
7190 


3280 
2U1I5 

612 
1873 

476 
1996 

1900 
2786 

442 
773 


; ; 

•319 

•i56 
•193 

•210 

■125 
•137 
•133 

•278 
•329 


'•81 

1-45 

8^51 
15^08 

8-01 
12-23 

7-87 
2500 
68-91 

2-n 

5-06 


3 

1 

v 

2 
5 
6 
4 

s' 

7 


5 
1 

•• 

2 

i 

6 

8 

4 

8 


1-8 

1-6 

2-4 
20 

2-8 
/to anv 
1 degree 

f to any 
(.degree 

f to any 
(degree 

1-6 


3 

y 
v 

4 

•• 

5 

7 


3-8 
100 

i-8 

9-7 
8^9 
8^6 
3-7 

30 
3-6 


Marble, average 

Granite, ditto . 

Purbeck stone . 

Portland ditto . 

Bristol ditto . . 

Jlillstone . . . . 

Pa viug stone. .' 

Craigleith ditto 

Grindstone . . . 

Chalk, Brit. . . 

Brick 

Coal, Scotch . . 
" Newcastle 
" Staffordsh. 
" Cannel . . 


2730 
2651 
2601 
2570 
2554 
2484 
2415 
2362 
2143 
2781 
2000 
1300 
1270 
1240 
1238 


170-00 
165-68 
162-56 
160-62 
159-62 
155-25 
15093 
147-62 
133-93 
173-81 
125-00 
8115 
79-37 
77-50 
77^37 


13 

13>^ 

!f 

14 

If 

29 
29 


9^25 1 
6-2 
9-0 1 

''.\ 

57 
5-0 
66 
0^5 
0^8 



Table containing the Weight of* Cokimns of Water, each one foot 
in length, and of Various Diameters, in lbs. avoirdupois. 



m^ 


■Weight. 


Diam. 


Weight. 


Diam. 


Weight. 


Diam. 


•Weight. 


Diam. 


Weight 


Diam. 


Weight. 


3 in. 


3-0672 1 


9 in. 


27-6120 


15 in. 


76-7004 


21 in. 


150-2376 


27 in. 


248-.5116 


33 in. 


371-2344 


/4 


3-3288 


yc 


28-3848 


/S 


77-9344 1 


Y^ 


152^1288 


li 


250-8180 


Y 


3740520 


1^ 


3.6000 


^4 


291672 


/4 


79-2792 


K 


153^9348 


¥- 


253-13.52 




376-8004 


a 


3-8820 


% 


29-9604 


% 


80-5836 i 


% 


1557396 




255-46".2 


' % 


379-4592 


>^ 


4-1748 


\/ 


30-7657 


yi 


81-9000 


^2 


1.57^5780 


x/ 


257-8003 


% 


382-5684 




4-4784 


% 


31-6524 


% 


83-2260 


% 


159^4152 


% 


260-1504 




385-4292 


i/ 


4-7928 


M 


32-4060 


V 


84-5628 


% 


161^2644 


sz 


262-5096 


3? 


388-2996 


% 


51180 


j^ 


33-2424 


% 


85-9104 


% 


1631220 


% 


264-8796 


34^n. 


391-1820 


4 in. 


5-4540 


10 in. 


340884 


16 in. 


87-2888 


22 in. 


164 9928 


28 in. 


267-2616 


394-0740 


% 


5-7996 


ya 


34-9464 




88-6.368 


Y, 


166^8732 


^a 


2696532 




396-9768 


}i 


6-1572 


4- 


35-8152 


y 


90-0168 


Y 


168^7632 


3i 


2720.544 


y. 


399-8928 




6-5244 




36-6936 


% 


91-4176 


% 


170-6652 


3t 


275-6672' 


'V 


402-8088 


L/ 


6-9024 


1 


37'5828 


/■a 


92-8080 


y^ 


172-5780 


1- 


276-8916 


f 


405-7500 


% 


7-2912 




33-4828 


% 


94-2192 


fi 


174-5004 


% 


279-:3252 


408-6948 


3^ 


7^6908 


3^ 


39-3936 


M 


95-6412 


176-4336 


i/ 


231-7708 


1/ 


411-4116 


yk 


8^1012 


% 


40-3152 


% 


970740 


'a 


178-3776 


% 


284-226M 


5a 


414-6180 


5 in. 


8^5212 


11 in. 


41-2476 


17 m. 


98-5176 


23 in. 


180-3324 


29 in. 


286 «920 


35 in. 


417-5952 




8^95.32 


% 


42-1908 


y^ 


99-9720 


% 


182-2980 


\a 


289-1688 




420-5844 


\x 


9^3948 




431436 


>4 


101-4.372 


>4 


184-2744 




291-6564 


.'i 


423-5S32 


J^ 


9^8484 


% 


14-1084 


% 


102-9120 


% 


186-2616 


% 


294-1548 


% 


426-5923 


i 


10-3126 


i< 


450S28 




104-3938 


fa 


188-2584 


'3 


296-5548 


y 


429-6120 


10-7856 


J^ 


46-0680 


y 


105-8952 


190-2672 




2991828 


% 


432-6432 


3^ 


11-2704 


M 


470610 


?€ 


107-4024 


^i 


192-2856 


5i 


301-7124 




435-6840 


?i 


11-7660 


1.4, 


48-0708 


% 


108-9204 


Va 


194-3184 


% 


304-2540 


y 


433 7368 


6 in. 


12-2712 


12 in. 


49-0334 


18 in. 


110-4492 


24 in. 


196-3548 


30 in. 


306-8052 


36 m. 


441-7992 


% 


12-7884 


y^ 


50-1168 


X 


111-9383 


Y 


198-4056 


Y 


309-3672 




447-9573 


J4 


133152 


y^ 


51-1548 


M 


113-5392 


Y 


200-4672 


Y 


311-9400 


'y 


454-1678 




13-8540 


% 


52-2048 


% 


115-0992 


Y 


203-5384 




314-.5224 


% 


460-4105 


\n 


14-4024 


^ 


53-2644 


■L/ 
/2 


116-6712 


K 


204-6216 


a 


317-1168 


A. 


466-6960 


s' 


14-9616 


% 


54-3318 




118-2528 


% 


206-7114 


/"a 


319-7220 




473-0210 


M 


155316 


?i 


.55-4760 


M 


119-8452 


% 


208-8192 


i 


322-.336S 


y 


479-3946 


T/' 


161124 


% 


.56-4804 


yk 


121-4484 


% 


210-9336 


324 9624 


4. 


485-8078 


7 in. 


167028 


13 in. 


57-6108 


19 in. 


123-0624 


25 in. 


213-0588 


3Un. 


327-6000 


492-2(i37 




173052 


3i 


58-7244 


y 


124-6872 


Y 


215-1948 


I'a 


.330-2472 




498-7621 


\x 


179172 


Y^ 


59-8476 




126-3228 


Y 


217-3416 


V 


332-9052 


y 


505-3032 


%. 


185412 


X 


60-9828 


y 


127-9680 


% 


219-4980 


% 


335.5728 


K 


511-9979 


x^ 


191748 




62-1276 


^2 


129-62.52 


yi 


221-6664 


i 


338-2.524 


39 in. 


518-41.32 


% 


19^8192 


% 


63-2832 


% 


131-5320 


% 


223-8441 


340-9428 




,525-1821 


%/ 


20-4744 


% 


64-4496 


M 


132-9696 


M 


226-0344 


343-6428 


v 


531-8936 


% 


211404 


% 


65-6268 


% 


134-6580 


% 


228-2340 


% 


346-3.536 


3/ 


.538-6478 


8 in. 


21-8172 


14 in. 


66-8148 


20 in. 


136-3562 


26 in. 


230-4t44 


32 in. 


349 0764 


40 m. 


545-4445 


% 


22-5036 


>3 


63-01.36 


>i 


138-0672 


ya 


232-6644 


K 


.3;)l-8088 


}4 


552-2S;W 


;| 


2.3-2020 


3-4 


69-2220 


'4 


139-7880 


>4 


234-8576 


Y 


,354-5520 


y 


559-1659 




23-9100 


X 


70-4424 


% 


141-5184 


% 


237-1404 




.3.57-.3048 


i5 
4l'm. 


566-0904 


% 


24-5288 


^ 


71-6724 


/8 


143-2608 




239-3928 


a 


360-0696 


5730577 


25-3.524 


% 


72-9120 


145-0128 


% 


241-6.572 


i/ 


362-8452 


y 

42 in. 


587-1199 


'A 


260988 


>*i 


711648 


?i 


146-7756 


% 


2^13-9312 


%, 


365-a304 


601-3526 


y» 


26 8500 


7? 


75-4272 


% 


148-5492 


Va 


246-2160 


% 


.368-4276 


50 in. 


799-2426 



2G 



402 



THE PRACTICAL MODEL CALCULATOR. 



Table containing the Weight of Square Bar Iron, from 1 to 10 feet 
in length, and from \ of an inch to 6 inches square. 



Inches 
square. 








LENGTH OF THE BARS 


IN f EET. 








1 foot. 


2 feet. 


3 feet. 


i feet. 


5 feet. 


6 feet. 


7 feet. 


8 feet. 


9 feet. 


10 feet. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


~Tb~ 


Lbs. 


Lbs. 


I 


0-2 


0-4 


0-6 


0-8 


1-1 


1-3 


1-5 


1-7 


1-9 


2-1 


1 


0-5 


1-0 


1-4 


1-9 


2-4 


2-9 


3-3 


3-8 


4-3 


4-8 




0-8 


1-7 


2-5 


3-4 


4-2 


5-1 


5-9 


6-8 


7-6 


8-5 


4 


1-3 


2-6 


4-0 


5-3 


6-6 


7-9 


9-2 


10-6 


11-0 


13-2 


3l 


1-9 


3-8 


5-7 


7-6 


9-5 


11-4 


13-3 


15-2 


17-1 


19-0 


|. 


2-6 


5-2 


7-8 


10-4 


12-9 


15-5 


18-1 


20-7 


28-3 


25-9 


1 in. 


3-4 


6-8 


10-1 


13-5 


16-9 


20-3 


23-7 


27-0 


30-4 


33-8 


H 


4-3 


8-6 


12-8 


17-1 


21-4 


25-7 


29-9 


34-2 


38-5 


42-8 




5-3 


10-6 


15-8 


21-1 


26-4 


31-7 


37-0 


42-2 


47-5 


52-8 


14 


6-4 


12-8 


19-2 


25-6 


32-0 


38-3 


44-7 


51-1 


57-5 


63-9 


11 


7-6 


15-2 


22-8 


30-4 


38-0 


45-6 


53-2 


60-8 


68-4 


76-0 


14 


8-9 


17-9 


26-8 


35-7 


44-6 


53-6 


62-5 


71-4 


80-3 


89-3 


H 


10-4 


20-7 


31-1 


41-4 


51-8 


62-1 


72-5 


82-8 


93-2 


103-5 


11-9 


23-8 


35-6 


47-5 


59-4 


71-3 


83-2 


95-1 


106-9 


118-8 


2 in. 


13-5 


27-0 


40-6 


54-1 


67-6 


81-1 


94-6 


108-2 


121-7 


135-2 


2^ 


15-3 


30-5 


45-8 


61-1 


76-3 


91-6 


106-8 


122-1 


137-4 


152-6 




17-1 


34-2 


51-3 


68-4 


85-6 


102-7 


119-8 


136-9 


154-0 


171-1 


2|. 


19-1 


38-1 


57-2 


76-3 


95-3 


114-4 


133-5 


152-5 


171-6 


190-7 


2^ 


21-1 


42-8 


63-4 


84-5 


105-6 


126-7 


147-8 


169-0 


190-1 


211-2 


^ 


23-3 


46-6 


69-9 


93-2 


11^-5 


139-8 


163-0 


186-3 


209-6 


232-9 


H 


25-6 


51-1 


76-7 


102-2 


127-8 


153-4 


178-9 


204-5 


230-0 


255-6 


H 


27-9 


55-9 


83-8 


111-8 


139-7 


167-6 


195-7 


223-5 


251-5 


279-4 


3 in. 


30-4 


60-8 


91-2 


121-7 


152-1 


182-5 


212-9 


243-3 


273-7 


304-2 


H 


33-0 


66-0 


99-0 


1320 


165-1 


198-1 


231-1 


264-] 


297-1 


330-1 


H 


35-7 


71-4 


107-1 


142-8 


178-5 


214-2 


249-9 


285-6 


321-3 


357-0 


3f 


38-5 


77-0 


115-5 


154-0 


192-5 


231-0 


269-5 


308-0 


346-5 


385-0 


H 


41-4 


82-8 


124-2 


165-6 


207-0 


248-4 


289-8 


331-3 


372-7 


414-1 


^ 


44-4 


88-8 


133-3 


177-7 


222-1 


266-5 


310-9 


355-3 


399-8 


444-2 


3f 


47-5 


95-1 


142-6 


190-1 


237-7 


285-2 


332-7 


380-3 


427-8 


475-3 


37 


50-8 


101-5 


152-3 


203-0 


253-8 


304-5 


355-3 


406-0 


456-8 


507-6 


4 in. 


54-1 


108-2 


162-3 


216-3 


270-4 


324-5 


378-6 


432-7 


486-8 


540-8 


4* 


57#5 


115-0 


172-6 
'l83-2 


230-1 


287-6 


345-1 


402-6 


460-1 


517-7 


575-2 


4i 


61-1 


122-1 


244-2 


305-3 


366-3 


427-4 


488-4 


549-5 


610-6 


4| 


64-7 


129-4 


194-1 


258-8 


323-5 


388-2 


452-9 


517-6 


582-3 


647-0 


4| 


68-4 


136-9 


205-3 


273-8 


342-2 


410-7 


479-1 


547-6 


616-0 


684-5 


44 


72-3 


144-6 


216-9 


289-2 


361-5 


433-8 


506-1 


578-4 


650-7 


723-1 


4^ 


76-3 


152-5 


228-8 


305-1 


381-3 


457-6 


533-8 


610-1 


686-4 


762-6 


4t 


80-3 


160-7 


241-0 


321-3 


401-7 


482-0 


562-3 


642-7 


723-0 


803-3 


Sin. 


84-5 


169-0 


263-4 


337-9 


422-4 


506-9 


591-4 


675-8 


760-3 


844-8 


5i 


93-2 


186-3 


279-5 


372-7 


465-8 


559-0 


652-2 


745-3 


838-5 


931-7 


5^ 


102-2 


204-5 


306-7 


409-0 


511-2 


613-4 


715-7 


817-9 


920-2 


1022-4 


6| 


111-8 


223-5 


335-3 


447-0 


558-8 


670-5 


782-3 


894-0 


1005-8 


1117-6 


6 in. 


121-7 


243-3 


365 


486-7 


608-3 


730-0 


841-6 


973-3 


1009-5 


1216-6 



Table of the Weight of a Square Foot of Sheet Iron in lbs. avoirdu- 
pois, the thickness being the number on the wire-gauge. JVo. 1 



IS ^ of an inch; No. 4, \; No. 11, \, ^c. 



No. on wire-gauge | 1 


2 


|3 


1 4 


|6] 6 


7 1 8 


9 


10 


11 


Pounds avoir [ 12-5 


12 


11 


10 


9] 8 


7-5 1 7 


6 


5-68 


5 


No. on wire-gauge | 12 


13 


14 


15 


16 j 17 


18 19 


20 


21 


22 


Pounds avoir | 4-62 


4-81 


* 


3 -go 


8 |2-.5 


2-18 1 1-93 


1-62 


1-5 


1-37 



SPECIFIC GRAVITY. 



403 



Table of the Weight of a Square Foot of Boiler Plate Iron, from 
^ to 1 inch thick, in lbs. avoirdupois. 



i 


-h 


ii A 


It 


tV 


^ 


1% 


t 


ii 


i 


ft 


^ 


if 


1 in. 


5 


7-6 


10 1 12-5 


15 


17-5 


20 


22-5 


25 


27-5 


30 


32-5 


35 


37-5 


40 



Table containing the Weight of Round Bar Iron, from 1 fo 10 feet 
in length, and from \ of an inch to 6 inches diameter. 



§1 


LEXGTH OF THE BARS IN FEET. 


1 foot. 


2 feet. 


3 feet. 


4 feet. 


5 feet. 


6 feet. 


7 feet. 


8 feet. 


9 feet. 


10 feet. 


Lbs. 

0-2 


"Lbs"" 


Lbs. 


Lbs. 


Lbs. 

0-8 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


\ 


0-3 


0-5 


0-7 


1-0 


1-2 


1-3 


1-5 


1-7 




0-4 


0-7 


1-1 


1-5 


1-9 


2-2 


2-6 


3-0 


3-4 


3-7 


i 


0-7 


1-3 


2-0 


2-7 


3-3 


4-0 


4-6 


5-3 


6-0 


6-6 ! 


h 


1-0 


2-1 


3-1 


4-2 


5-2 


6-3 


7-3 


8-3 


9-4 


10-4 


\ 


1-5 


3-0 


4-5 


6-0 


7-5 


9-0 


10-5 


11-9 


13-4 


14-9 


i 


2-0 


41 


6-1 


8-1 


10-2 


12-2 


14-2 


16-3 


18-3 


20-3 


1 in. 


2-7 


5-3 


8-0 


10-6 


13-3 


15-9 


18-6 


21-2 


23-9 


26-5 ! 


n 


3-4 


6-7 


10-1 


13-4 


16-8 


20-2 


23-5 


26-9 


30-2 


33-6 


H 


4-2 


8-3 


12-5 


16-7 


20-9 


25-0 


29-2 


33-4 


37-5 


41-7 


If 


5-0 


10-0 


15-1 


20-1 


25-1 


30-1 


35-1 


40-2 


45-2 


50-2 


u 


6-0 


11-9 


17-9 


23-9 


29-9 


35-8 


41-8 


47-8 


53-7 


59-7 


ll- 


7-0 


14-0 


21-0 


28-0 


35-1 


42-1 


49-1 


56-1 


63-1 


70-1 


If 


8-1 


16-3 


24-4 


32-5 


40-6 


48-8 


56-9 


65-0 


73-2 


81-3 


If 


9-3 


18-7 


28-0 


37-3 


46-7 


56-0 


65-3 


74-7 


84-0 


93-3 


2 in. 


10-6 


21-2 


31-8 


42-6 


53-1 


63-7 


74-3 


84-9 


95-5 


106-2 


2i 


12-0 


24-0 


36-0 


48-0 


59-9 


71-9 


83-9 


9,5-9 


107-9 


119-9 


2i 


13-4 


26-9 


40-3 


53-8 


67-2 


80-6 


94-1 


107-5 


121-0 


134-4 


2| 


15-0 


30-0 


44-9 


60-0 


74-9 


89-9 


104-8 


119-8 


134-8 


149-8 


2| 


16-7 


33-4 


50-1 


66-8 


83-5 


100-1 


116-8 


183-6 


150-2 


166-9 


2| 


18-3 


36-6 


54-9 


73-2 


91-5 


109-8 


128-1 


146-3 


164-6 


182-9 


2| 


20-1 


40-2 


60-2 


80-3 


100-4 


120-5 


140-5 


160-6 


180-7 


200-8 


H 


21-9 


43-9 


65-8 


87-8 


109-7 


131-7 


153-6 


175-6 


197-5 


219-4 


3 in. 


23-9 


47-8 


71-7 


95-6 


119-4 


143-3 


167-2 


191-1 


215-0 


238-9 


3^ 


25-9 


51-9 


77-8 


103-7 


129-6 


155-6 


181-5 


207-4 


233-3 


259-3 


3- 


28-0 


56-1 


84-1 


112-2 


140-2 


168-2 


196-3 


224-3 


253-4 


280-4 


3- 


30-2 


60-5 


90-7 


121-0 


151-2 


181-4 


211-7 


241-9 


272-2 


302-4 


3 ' 


32-5 


65-0 


97-5 


130-0 


162-6 


195-1 


227-6 


260-1 


292-6 


325-1 


H 


34-9 


69-8 


104-7 


139-5 


174-4 


209-3 


244-2 


279-1 


314-0 


348-9 


3| 


37-3 


74-7 


112-0 


149-3 


186-7 


224-0 


261-3 


298-7 


336-0 


373-3 


3i 


39-9 


79-7 


119-6 


159-5 


199-3 


239-2 


279-0 


318-9 


358-8 


398-6 


4 in. 


42-5 


84-9 


127-4 


169-9 


212-3 


254-8 


297-2 


339-7 


382-2 


424-6 


4^ 


45-2 


90-3 


135-5 


180-7 


225-9 


271-0 


316-2 


361-4 


406-6 


451-7 


41 


48-0 


95-9 


143-9 


191-8 


239-8 


287-7 


335-7 


383-6 


431-6 


479-5 


4| 


50-8 


101-6 


152-4 


203-3 


254-1 


304-9 


355-7 


406-5 


457-3 


508 2 


4^ 

^ 


53-8 


107-5 


161-3 


215-0 


268-8 


322-6 


376-3 


430-1 


483-a 


537-6 


56-8 


113-6 


170-4 


227-2 


283-9 


340-7 


397-5 


454-3 


511-1 


567-9 


^. 


60 


119-8 


179-7 


239-6 


299-5 


359-4 


419-3 


479-2 


539-1 


599-0 


^ 


63-1 


126-2 


189-3 


252-4 


315-5 


378-6 


441-7 


504-8 


567-8 


630-9 


5 in. 


66-8 


133-5 


200-3 


267-0 


333-8 


400-5 


467-3 


534-0 


600-8 


667-5 


5i 


73-2 


146-3 


219-5 


292-7 


365-9 


439-0 


512-2 


585-4 


658-5 


731-7 


5J 


80-3 


160-6 


240-9 


321-2 


401-5 


481-8 


562-1 


642-4 


722-7 


803-0 


5| 


87-8 


175-6 


263-3 


351-1 


438-9 


526-7 


614-4 


702-2 


790-0 


877-8 


6 in. 


95-6 


191-1 


286-7 


382-2 


477-8 


573-3 


668-9 


764-4 


860-0 


965-5 



Table of the Weight of Oast Iron Plates, per Superficial Foot, from 
one-eighth of an inch to one inch thick. 



% inch. 1,1 inch. 


% inch. 


}.< inch. 


y^ inch. 


% inch. 


%inch. linch. 


lbs. OS. lbs. oz. 

4 13| 9 10| 


lbs. oz. 

14 8 


lbs. cz. 

19 5f 


lbs. OZ. 

24 2| 


lbs. OZ. 

29 


lbs. oz. 

33 13| 


lbs. oz. 

38 lOf 



4C4 



THE PRACTICAL MODEL CALCULATOR. 



Table 


containing the Weight of Cast Iron Pipes, J 


foot in 


length. 


'- o ^ 

m 








THICKNESS 


IN INCHES. 






^ 


1 


J 


t 


f 


i 


1 inch. 


n 


11- 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


ij 


6-9 


9-9 














2 


8-8 


12-3 


16 -i 


2()-3 










2h 


10-6 


14-7 


19-2 


23-9 










3 


12-4 


17-2 


22-2 


27-6 


33-3 


39-3 


45-6 




3* 


14-2 


19-6 


25-3 


31-3 


37-6 


44-2 


51-1 




4 


16-8 


22-1 


28-4 


35-0 


41-9 


49-1 


56-6 


64-4 


4* 


18-0 


24-5 


31-4 


38-7 


46-2 


54-0 


62-1 


70-6 


5 


19-8 


. 27-0 


34-5 


42-3 


50-5 


58-9 


67-6 


76-7 


5^ 


21-6 


29-5 


37-6 


46-0 


54-8 


63-8 


73-2 


82-8 


6 


23-5 


31-9 


40-7 


49-7 


59-1 


68-7 


78-7 


88-8 


6* 


25-3 


34-4 


43-7 


53-4 


63-4 


73-4 


84-2 


95-1 


7 


27-2 


36-8 


46-8 


56-8 


67-7 


78-5 


89-7 


101-2 


7h 


29-0 


39-1 


49-9 


60-7 


72-0 


83-5 


95-3 


107-4 


8 


80-8 


41-7 


52-9 


64-4 


76-2 


88-4 


100-8 


113-5 


8^ 


32-9 


44-4 


56-2 


68-3 


80-8 


93-5 


106-5 


119-9 


9 


34-5 


46-6 


59-1 


71-8 


84-8 


98-2 


111-8 


125-8 


9^ 


36-3 


49-1 


624 


75-5 


89-1 


103-1 


117-4 


131-9 


10 


38-2 


51-5 


65-2 


79-2 


93-4 


108-0 


122-8 


138-1 


10* 





54-0 


68-2 


82-8 


97-7 


112-9 


128-4 


144-2 


11 




56-4 


71-3 


86-5 


102-0 


117-8 


133-9 


150-3 


llj 




58-9 


74-3 


90-1 


106-3 


122-7 


139-4 


156-4 


12 




61-3 


77-4 


93-6 


110-6 


127-6 


145-0 


162-6 


13 







82-7 


101-2 


118-2 


137-4 


154-1 


173-5 


14 







89-5 


108-2 


126-5 


146-2 


165-3 


185-2 


15 






95-2 


115-7 


135-3 


156-2 


176-2 


198-1 


16 








123-3 


143-1 


166-1 


187-5 


211-3 


17 








130-2 


152-5 


178-5 


198-2 


228-4 


18 









137-0 


161-2 


185-3 


209-1 


235-6 


19 











169-2 


195-7 


222-3 


247-1 


20 










178-1 


205-2 


233-2 


259-0 


21 













214-1 


243-5 


278-2 


22 













223-0 


254-8 


285-4 


23 













233-4 


265-5 


298-3 


24 













245-2 


277-5 


310-6 



Table containing the Weight of Solid Cylinders of Cast Iron, one 
foot in length,_ and from f of an inch to 14 inches diameter. 



Diameter in 


Weight ia 


Diameter in 


Weight in 


Diameter in 


Weight in 


Diameter in 


Weight in 


Inches. 


Lbs. 


Inches. 


Lbs. 


Inches. 


Inches. 


Lbs. , 


f 


1-39 


2, 


20-48 


4t 


58-72 


7| 


148-87 


i 


1-88 


3 in. 


22-35 


5 in. 


61-96 


8 in. 


158-68 


lin. 


2-47 


H 


24-20 


5* 


64-66 


8i 


168-15 


1^ 


3-18 


3i 


26-18 


5i 


68-31 


8^ 


179-08 


1^ 


3-87 


H 


28-23 


5| 


71-00 


8f 


189-00 


If 


4-68 


3* 


30-36 


^ 


74-98 


9 in. 


200-77 


1|. 


5-57 


H 


32-57 


^ 


78-65 


^4 


211-12 


If 


6-54 


3| 


34-85 


5f 


81-95 


n 


223-70 


1|" 


7-59 


H 


37-21 


4 


85-81 


9f 


235-31 


8-71 


4 in. 


39-66 


6 in. 


89-23 


10 in. 


247-87 


2in. 


9-91 


4* 


41-80 


6i 


96-82 


m 


273-27 


2i 


11-19 


4i 


44-77 


6* 


104-72 


11 in. 


299-92 


2i 


12-54 


^8 


47-00 


6f 


112-93 


IIJ 


327-81 


2| 


13-98 


4j 


50-19 


7 in. 


121-45 


12 in. 


856-93 


2^ 


15-49 


4f 


52-71 


7i 


130-28 


13 


418-90 


2| 


17-08 


4| 


55-92 


7J 


139-42 


14 


485-88 


2| 


18-74 















SPECIFIC GRAVITY. 



405 



Table containing the Weight of a Square Foot of Copper and 
Lead, in lbs. avoirdupois, from ^to ^ an inch in thickness, ad- 
vancing hy ^. 



Thickness. 


Copper. 


Lead. 


A 


1-45 


1-85 


h 


2-90 


3-70 


^ 


4-35 


5-54 


I 


5-80 


7-39 


i + A 


7-26 


9-24 


i +^ 


8-71 


11-08 


i +^ 


10-16 


12-93 


i 


11-61 


14-77 


I +i^ 


13-07 


16-62 


I +1^ 


14-52 


18-47 


\ +^ 


15-97 


20-31 


f 


17-41 


22-16 


f + A 


18-87 


24-00 


f +1^6 


20-32 


25-85 


1 +^ 


21-77 


27-70 


i 


23-22 


29-55 



Table for finding the Weight of Malleable Iron, Copper, and Lead 
Pipes, 12 inches long, of various thicknesses, and any diameter 
required. • 



Thickness. 


Malleable Iron. 


Copper. 


Lead. 


^ of an inch. 


•104 


-121 


•1539 


1^6 


-208 


•2419 


•3078 


h 


-3108 


•3628 


-4616 


\ 


-414 


•4838 


•6155 


\ 4- A 


•518 


•6047 


•7694 


\ ^ h 


-621 


•7258 


•9232 


\ + h 


-725 


•8466 


1-0771 


\ 


-828 


•9678 


1-231 



Rule. — Multiply the circumference of the pipe in inches by the 
numbers opposite the thickness required, and by the length in feet ; 
the product will be the weight in avoirdupois lbs. nearly. 

Required the weight of a copper pipe 12 feet long, 15 inches in 



circumference, \ + 



i^g of an inch in thickness. 



•7258 X 15 = 10-817 x 12 = 130-644 lbs. nearly. 



Table of the Weight of a Square Foot of Millboard in lbs. avoirdupois 



Thickness in jnches 


i 


1^ 


\ ^ 


f 


Weiffht in lbs 


•688 


1-032 


1-376 


1-72 2-064 






1 



406 



THE PRACTICAL MODEL CALCULATOR. 



Table containing the Weight of Wrought Iron Bars 12 inches long 

in lbs. avoir ' 



Inch. 


Round. 


Square. 


Inch. 


Round. 


Square. 


1 


•163 


•208 


^ 


16-32 


20-80 


3 


•367 


•467 


2| 


18-00 


22-89 


X 


•653 


•830 


2| 


19-76 


25-12 


h 


1-02 


1-30 


2| 


21-59 


27-46 


|. 


1^47 


1-87 


3 


23-52 


29-92 


7 


2-00 


2-55 


H 


27-60 


35-12 




2-61 


3-32 


H 


32-00 


40-80 


H 


3-31 


4-21 


3| 


36^72 


46-72 




4-08 


5-20 


4 


41^76 


53-12 


i-j. 


4-94 


6^28 


4i 


47^25 


60-00 


li 


5-88 


7-48 


4J 


52-93 


67-24 


i-| 


6-90 


8-78 


4f 


^8-92 


74-95 


la 


8-00 


10-20 


5 


65-28 


83-20 


17 


9-18 


11-68 


5i 


72-00 


91-56 


2 


10^44 


13-28 


5J 


79-04 


100-48 


2i 


11-80 


15-00 


5| 


86-36 


109-82 


2i 


13-23 


16-81 


6 


94-08 


119-68 


2| 


14-73 


18-74 


7 


128-00 


163-20 



Table of the Proportional Dimensions of 6-sided Nuts for Bolts from 
^ to 2J inches diameter. 



Diameter of bolts 


i 


fh 


f 


1 

4 


1 


1 


li 


11 


Breadth of nuts 


« 


ilil 


lA 


If 


lA 


If 


IH 


2| 


Breadth over the angles 


1 


If 


n 


If 


lA 


1« 


2 


2i 


2A 


Thickness 


aU 


A 


3 

4 


7 
8 


1 


li 


li 


lA 


Diameter of bolts 


If 


li 


If 


If 


If 


2 


2i 


2i 




" Breadth of nuts 


2A 


2| 


2tt 2| 


3A 


3i 3| 


4 










Breadth over the angles 


2« 


2| 


8i 


3^ 


3i 


3| 


4A 


4f 




Thickness 


lA 


m 


1« 


2 


2i 


2i 


2i 


2| 





Table of the Specific G-ravity of Water at different temperatures, 
that at 62° heing taken as unity. 



70° F. 


•99913 


52° F. 


1-00076 


68 


-99936 


50 


1-00087 


66 


•99958 


48 


1^00095 


64 


•99980 


46 


1-00102 


62 


1- 


44 


1-00107 


58 


1-00035 


42 


1-00111 


56 


1-00050 


40 


1-00113 


54 


1-00064 


38 


1-00115 



I 



The difference of temperatures between 62° and 39°'2, ivhere 
water attains its greatest density, will vary the bulk of a gallon 
rather less than the third of a cubic inch. 



SPECIFIC GRAVITY. 



40: 



Table of the Weight of Cast Iron Balls in pounds avoirdupois, 
from 1 to 12 inches diameter^ advancing hy an eighth. 



Inches. 


Lbs. 


Inches. 


Lbs. 


Inches. 


Lbs. 




•14 


4f 


14-76 


8^ 


84-56 


H 


•20 


4 


15-95 


8f 


88-34 




•27 


5 


17-12 


8f 


92-24 


i-| 


•37 


^ 


18-54 


8-1 


96-26 


i| 


•47 


^4 


19-93 


9 


100^39 


•59 


5| 


21-39 


9* 


104-62 


If 


•74 


H 


22-91 


9? 


108-98 


1^ 


•91 


4 


24-51 


9| 


, 113-46 


2 


1^10 


5| 


26-18 


9? 


118-06 


2i 


r32 


■ H 


27-91 


9^ 


122-77 


2| 


1-57 


6 


29-72 


9f . 


127-63 




1-84 


6^ 


31-64 


91 


132-60 


2I 


2-15 


61 


33-62 


10 


137-71 


24 


2-49 


6f 


35-67 


lOi 


142-91 


2f 


2-86 


^ 


37-80 


lOj 


148-28 


2i 


3-27 


6f 


40-10 


10^ 


153-78 


3 


3^72 


6f 


42-35 


lOi 


159-40 


H 


4^20 


6i 


44-74 


lOf 


165-16 


31 


4^72 


7 


47-21 


lOf 


171-05 


H 


5^29 


7i 


49-79 


lOi 


177-10 


H 


5-80 


7i 


52-47 


11 


183^29 


3f 


6-56 


7| 


55-23 


lu 


189-60 


3| 


7-26 


7i 


58-06 


Hi 


196-10 


H 


8-01 


7| 


60-04 


11^ 


202-67 


4 


8-81 


7f 


6409 


11? 


209-43 


^ 


9-67 


"i" 


67-25 


11^ 


216-32 


4i 


10^57 


8 


70-49 


111 


223-40 


^ 


11-53 


8i 


73 '85 


Hi 


230-57 


^ 


12-55 


81 


77-32 


12 


237-94 


^ 


13-62 


8| 


80-88 


1 





Table of the Weight of Flat Bar Iron, 12 inches long, in lbs. 

avoirdupois. 



Thickness. 


i 


1% 


i 


1 


i 


1 


3 
■1 


i 


1 inch. 




1 


-21 


•31 


•42 


-63 














i 


•31 


•47 


•63 


•94 


1-26 


1-57 










1 


•42 


•63 


•84 


1-26 


1-68 


2-10 


2-52 


*2-94 






H 


-52 


-78 


1-05 


1-57 


2-10 


2-62 


3-15 


3-67 


4-20 




If 


-57 


•86 


1-18 


1-73 


2-31 


2-88 


3-46 


4-04 


4-62 




u 


•63 


•94 


1^26 


1-89 


2-52 


3-15 


3-78 


4-41 


5-04 


t^ 


If 


•73 


1-10 


1-47 


2-20 


2-94 


3-67 


4-41 


5-14 


5-87 


-i 


2 


•84 


1^26 


1^68 


2-52 


3-36 


4-20 


5-06 


5-88 


6^72 




2^ 


•96 


1-41 


1-89 


2-83 


3-78 


4-72 


5-66 


6-61 


7-56 


.9 


2* 


1-05 


1-57 


2^10 


3-15 


4-20 


5-25 


6-30 


7-35 


8-40 


f. 


23. 


1-15 


1-73 


2-31 


3-46 


4-62 


5-77 


6-93 


8-08 


9-24 




3 


1-26 


1-89 


2-52 


3-78 


5-04 


6-30 


7-56 


8-82 


10-08 


Z 


31- 


1-36 


2-04 


2-73 


4-09 


5-46 


6-82 


8-19 


9-55 


10-92 


w 


3;^- 


1-47 


2-20 


2-94 


4-41 


5-88 


7-35 


8-82 


10-29 


11-76 




^ 


1-57 


2-36 


3-15 


4-72 


6-30 


7-87 


9-45 


11-02 


12-60 




4 


1-68 


2-52 


3-36 


5-04 


6-72 


8^40 


10-08 


11 76 


13-44 




n 


1-89 


2-83 


3-73 


5-67 


7-56 


9-45^ 


11-34 


13-23 


15^12 




5 


2-10 


3-15 


4-12 


6-30 


8-40 


10-50 


12-60 


16-70 


17-80 




6 


2-52 


3-78 


5-04 


7-56 


10-08 


12-60 


15-12 


17-64 


20-16 



Weight of a copper rod 12 inches long and 1 inch diameter = 3-039 lbs. 
Weight of a brass rod 12 inches long and 1 inch diameter = 2-86 lbs. 



408 



THE PRACTICAL MODEL CALCULATOR. 



Brass. — Weight of a Lineal Foot of Round and Square. 



Diameter. 


Weight of 
round. 


Weight of 
square. 


Diameter. 


Weight of 
round. 


Weight of 
square. 


Inches. 


Lbs. 


Lbs. 


Inches. 


Lbs. 


Lbs. 


1 


•17 


•22 


If 


8-66 


11-03 


|. 


•39 


•50 


H 


9-95 


12-66 


1 


•70 


•90 


2 


11-32 


14-41 


|. 


1-10 


1^40 


2i 


12-78 


16-27 


.| 


1-59 


2-02 


2i 


14-32 


18-24 


7 


2-16 


2^75 


2t 


15-96 


20-32 


1 


2-83 


3-60 


2J 


17-68 


22-53 


H 


3^58 


4-56 


2f 


19-50 


24-83 


1? 


«t-42 


6^63 


21 


21-40 


27-25 


If 


5-35 


6-81 


2i 


23-39 


29-78 


1? 


6^36 


8-00 


3 


25-47 


32^43 


i| 


7-47 


9^51 









Steel. — Weight of One Foot of Round Steel. 



Diameter in 

inches and 

parts. 


•167 


1 
•376 


•669 


1 
1^04 


1 
1-5 


i 
2-05 


1 
2-67 


3-38 


4-18 


If 
5-06 


6-02 


1| 
7-07 


If 

8-2 


It 


2 


Weight in 
libs, and deci- 
1 mal parts. 


.)-a 


11-71 



Tables of the Weights of Eolled Iron, 

Per lineal foot, of various sections, illustrated in tlie accompanying cuts, viz. 
Parallel Angle Iron, equal and unequal sides ; Taper Angle Iron; Parallel T 
Iron, equal and unequal depth and width; Taper T Iron; Sash Iron; and Per- 
manent and Temporary Rails. 

Table I. — Parallel Angle Iron, of equal sides. (Fig. 1.) 



Length of sides 


Uniform thickness 


A\ eight of one 
lineal foot 


AB, in inches. 


throughout. 


in lbs. 


Inches. 


1 -.Bi. 




3 


1 


8-0 


2| 


1 


7-0 


2i 


1 


5-75 


2i 


6-16ths 


4-5 


2 


ifull 


3-75 


If 


\ 


3-0 


1^ 


i 

4 


2-5 


If 


No. 6 wire-gauge 


1-75 


li 


8 


1-5 


H 


9 


1-25 




10 


1-0 


i 


10 


•875 




11 


•625 


|. 


11 


•568 


i 


12 


■5 



Fig. 1. 



f 



u _ V 



SPECIFIC GRAVITY. 



409 



Table II. — Parallel Angle Iron^ of unequal sides. (Fig. 



Length of side A, 
in inches. 


Length of side B, 
in inches. 


Uniform 

thickness 

throughout. 


Weight of one 

lineal foot 

in lbs. 


Inches. 

f 

3 

f 
f 

l| 


Inches. 

5 
5 
4 
4 
4 
4 
3 

f 

2 


Inches. 
1 

5-16ths 

5-16ths 

i 

3-16ths 


9-75 

8-75 

7-5 

6-75 

5-75 

5-5 

4-75 

3 •875 

2-875 

2-25 



Fig. 2. 
— A 



^m^^ww 



Table III. — Tajper Angle Iron, of equal sides. (Fig. 3.) 



Length of sides, 


Thickness of 


Thickness of root 


Weight of one 


A A, in inches. 


edges at B. 


ate. 


in lbs. 


Inches. 


Inches. 


Inches. 




4 


J 


1 


14-0 


3 


1 


|. 


10-375 


2f 


7-16ths 


g-ieths 


8-25 


^ 


1 


1 


6-5 


2-1- 


5-16ths, full 


7-16tlis 


5-0 


2 


Hiin 


5-16ths, full 


3-875 


If 


i 


5-16th3 


3-25 


IJ 


\ bare 


5-16tlis, bare 


2-625 




Table IV. — Parallel J Iron, of unequal width and depth. (Fig. 4. 



Fig. 4. 



Width of 
top table 


Total 
depth 
B, in 
inches. 


Uniform thick- 


Uniform 


Weight of one 


ness of top 


thickness of 


lineal foot 


inches. 


table C. 


ribD. 


in lbs. 


Inches. 


Inches. 


Inches. 


Inches. 




5 


6 


^ 


} 


15-75 


4* 


H 


^ 


9.16tlis 


13-25 


4 


3 


|. 


|. 


8-875 


3-] 


3 


f 


1 


8-25 


H 


4 


^ 


t 


12-5 


2J 


3 


f 


7-0 


2i 


2 


5-16ths 


ffull 


4-5 


2 


n 


5-16ths 


5-l(5ths 


4-0 


If 


2 


4 


4' « 


3-125 


Ik 


2 






2-875 


1]- 


u 


^ 


-- 


2-375 


1 


1]- 


3-16th3 


3-16ths 


1-5 


f 


1 


3-16ths 


3-16ths 


1-125 



imi^^m^^^^^^^^ ! 



410 



THE PRACTICAL MODEL CALCULATOR. 



Table V. — Parallel J Iron, of equal depth and width. (Fig. 5.) 



Width of top 


Uniform 


Weight of one 


table, and total 


thickness 


lineal foot 


depth A A. 


throughout. 


in lbs. 


Inches. 


Inches. 




6 


J 




5 


7-16tlis 


13-75 


4 


1 


9-75 


H 




8-5 


3 


-| 


7-5 


2^ 


5-16ths 


4-625 


4 


6-16ths 


4-5 


2 


5-16ths 


3-75 


If 


J 


3-0 


1^ 


^ 


2-25 


H 


i. 


1-75 


1 


3-16ths 


1-0 


1 


i 


•725 


f 


i 


-625 



Fig. 5. 
r*^ -A ^ 

J 



Table VI. — Taioer T Iron. 



Width of 

top table 

A, in 

inches. 


Total 
depth 
B, in 
inches. 


Thickness of 

top table at 

root C. 


Thickness of 

top table at 

edges U. 


Uniform 

thickness of 

rib E. 


1 

Weight of one 

lineal foot 

in lbs. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 




3 


3^ 


\ 


1 


7-16tlis 


8-0 


3 


2-1 


7-16tlis 


|. 


\ 


8-0 


2^ 


3 


7-16tlis 


5-16tlis 


5-16tlis 


5-25 


2 
2 


2* 
U 


1 
ffull 


S-ieths 


1 
1 


6-5 
3-5 


2 


H 


5-16ths 


\ 


\ 


2-875 



(Fig. 6.) 

Fig. 6. 

r A ^ 

F 



Table Nil.— 8a%}i Iron. (Fig. 7.) 



Total 


Depth of 


Width at 


Greatest 


Weight of one 


depth A. 


rebate B. 


edge C. 


'ividth D. 


in lbs. 


In lies. 


Inches. 




Inches. 




2 


1 


No. 9 wire-gauge 


5-8ths 


1-75 


If 


f 


7 


9-16ths 


1-625 


1? 


f 


6 


9-16ths 


1-25 


If 


1 


10 


9-16ths 


1-125 


U 


1 


10 


9-16tlis 


1-0 


1 


\ 


\ 


h 


•75 



Table VIII. — Raih equal top and bottom 
Tables. (Fig. 8.) 



Depth A, in 
inches. 


Width across top 

and bottom BB, 

in inches. 


Thickness of 
ribC. 


Weight of one 

lineal foot 

in lbs. 


Inches. 

5 

^ 

4i 


Inches. 

2| 

^ 


Inches. 
f 

1 


25-0 

23-83 

21-66 




Fig. 8. 




SPECIFIC GRAVITY. 



411 



Table IX. — Temporary Rails. (Fig. 9.] 



Top width 
A, in 


Rib Width 
B, in 


Bed width 
C, in 


Total depth 
D, in 


Thickness 
of bed E. 


Weight of one 
lineal foot 


inches. 


inches. 


inches. 


inches. 


in lbs. 


Inches. 


Inches. 


Inches. 


InchA. 


Inches. 




H 


1 


3 


2 


7-16ths 


9-0 


If 




3 


n 


^ 


12-0 


U 


1 


4 


8 


^ 


16-0 


2 


1 


4 


3 


k 


17-83 




Table of Natural Sines, Co-sines, Tangents, Co-tangents, Secants, 
and Co-secants, to every degree of the Quadrant. 



Deg. 


Sines. 


Co-sines. 


Tangents. 


Co-tangents. 


Secants. 


Co-secants. 


Degree. 





•00000 


1-00000 


•00000 


Infinite. 


1-00000 


Infinite. 


90 


1 


•01745 


•99985 


•01746 


57-2900 


1-00015 


57-2987 


89 


2 


•03490 


-99989 


•08492 


28-6868 


1-00061 


28-6537 


88 


3 


•05234 


•99868 


•05241 


19^0811 


1-00137 


19-1073 


87 


4 


•06976 


•99756 


•06993 


14^3007 


1^00244 


14-3356 


86 


5 


•08716 


•99619 


•08749 


11-4301 


1 •00882 


11-4737 


85 


6 


•10453 


•99452 


•10510 


9-51236 


1-00551 


9-56677 


84 


7 


•12187 


•99255 


-12278 


8^14435 


r00751 


8-20551 


88 


8 


•13917 


•99027 


•14054 


7-11537 


1 •00983 


7-18530 


82- 


9 


•15643 


•98769 


•15838 


6^31875 


1^01246 


6-89245 


81 


10 


•17365 


•98481 


•17638 


5^67128 


1-01543 


5^75877 


80 


11 


•19081 


•98163 


•19438 


5-14455 


1-01872 


5-24084 


79 


12 


•20791 


•97815 


•21256 


4^70463 


1-02234 


4-80973 


78 


13 


•22495 


•97437 


•23087 


4^88148 


1-02680 


4-44541 


77 


14 


•24192 


•97030 


•24933 


4-01078 


1-08061 


4-18356 


76 


15 


•25882 


•96598 


•26795 


3-78205 


1-03528 


8-86370 


75 


16 


•27564 


•96126 


•28675 


3-48741 


1-04030 


3-62796 


74 


17 


•29237 


•95630 


•30573 


3-27085 


1-04569 


3-42030 


73 


18 


•30902 


•95106 


-82492 


3^07768 


1-05146 


3-23607 


72 


19 


•32557 


•94552 


-34433 


2-90421 


1-05762 


3-07155 


71 


20 


•84202 


-93969 


-86897 


2^74748 


1-06418 


2-92380 


70 


21 


■35837 


-93358 


•88386 


2-60509 


1-07114 


2-79043 


69 


22 


•37461 


•92718 


■40403 


2-47509 


1-07858 


2-66947 


68 


23 


•39073 


•92050 


-42447 


2-35585 


1-08686 


2-55930 


67 


24 


•40674 


•91355 


•44523 


2^24004 


1-09464 


2-45859 


66 


25 


•42262 


-90681 


•46631 


2-14451 


1-10888 


2-86620 


65 


26 


•43837 


-89879 


•48773 


2-05080 


1-11260 


2-28117 


64 


27 


•45399 


•89101 


•50952 


1^96261 


1-12283 


2-20869 


63 


28 


•46947 


•88295 


•58171 


1^88073 


1-18257 


2^18005 


62 


29 


•48481 


•87462 


-55481 


1-80405 


1-14835 


2-06266 


61 


30 


•50000 


-86603 


-57785 


1-73205 


1-15470 


2^00000 


60 


31 


•51504 


-85717 


-60086 


1-66428 


1-16663 


1^94160 


59 


32 


•52992 


•84805 


•62487 


1-60033 


1-17918 . 


1^88708 


58 


83 


•54464 


•83867 


-64941 


1-53986 


1-19236 


1^83608 


57 


34 


•55919 


-82904 


■67451 


1-48256 


1-20622 


1^78829 


66 


35 


•57358 


-81915 


•70021 


1-42815 


1-22077 


1^74345 


55 


36 


•58778 


-80902 


•72654 


1-37688 


1-23607 


r70130 


54 


37 


•60181 


•79863 


•75355 


1-82704 


1-25214 


1-66164 


53 


38 


•61566 


■78801 


•78129 


1-27994 


1-26902 


1-62427 


52 


39 


•62932 


•77715 


•80978 


1-23490 


1-28676 


1-58902 


51 


40 


•64279 


•76604 


■83910 


M9175 


1-30541 


1-55572 


50 


41 


•65606 


•75471 


•86929 


1-15037 


1-32511 


1-52425 


49 


42 


•66913 


•74314 


■90040 


1-11061 


1-34561 


1-49448 


48 


43 


•68200 


•73185 


•93251 


1-07237 


1-36706 


1-46628 


47 


44 


•69466 


•71934 


-96569 


1-03553 


1-39012 


1-43956 


46 


45 


•70711 


•70711 


1-00000 


1-00000 


1-41421 


1-41421 


45 


Deg. 


Co-sines. 


Sines. 


Co-tangents. 


Tangents. | 


Co-secants. 


Secants. 


Degree. 



412 



MOMENT OF INERTIA. 

CORDS, KNOTS, NODES, CHAIN-BRIDGE. — ANGULAR VELOCITY. — RADIUS 
OF GYRATION. 

1. If the cord q NB, be fixed at the extremity B, and stretched 
by a weight of 500 lbs. at the extremity q, and the middle knot or 
node N, by a force of 255 lbs. pulling upwards, under an angle 
« N 5 of 54° ; what is the tension and position of NB. 




Angle ^Nr = 180° - angle gNP; and90°-aN5 = 5N(? = 
^Nr = 36°; cos. 36° = -80902. 

v/5002 ^ 2552 - 2 X 255 X 500 x cos. 36° = 329-7 lbs., the 
magnitude of tj^e tension. 

——J- = -891386 = sine of angles 5 N s, or angle BN r = 

63° 2^ 

2. Between the points A and B, a cord 10 feet in length is 
stretched by a weight W of 500 lbs. suspended to it by a ring ; 
the horizontal distance AE = Q-Q feet, and the vertical distance 
BE = 3-2 feet; required the position of the ring C, the tensions, 
and directions of the rope. 

The tensions of the cords AC, CB are equal, and angle AC h = 
angle 5 CB. 



MOMENT OF INERTIA. 



413 



1 

I 



""y>j 



w 



AD = AC + CB = 10 feet. 
^(102 - Q'Q^) = 7-5126 = ED; BD = 7-5126 - 3-2 = 4-3126 



J)n 



4-3126 

2 



2-1563; 7-5126 : 2-1563 : : 10 : 



21-563 
7-5126 



2-87 = CD = CB ; and CA 
Bn 



10 



2-87 = 7-13. 



_ 2-1563 
-g ^ = cosine bCB = ^.g^ ' = -75132. 



W 



-.^5CB = 41oi8-2^^^;-4^^, = 



500 



1-50264 



332-7 lbs.. 



tiie tension on the cord CB, which is equal to the tension 
on AC. 



L 



414 



THE PRACTICAL MODEL CALCULATOR. 



3. Let 500,000 lbs. be the whole 
weight on a chain-bridge whose span 
AB = 400 feet, and height of the arc 
CD = 40 feet ; required the tensions 
and other circumstances respecting the 
chains. 

The tangent of the angles of incli- 
nation of the ends of the chain is 
equal 

40 X 2 
■ 200 ~ '40000, the angle answer- 
ing to this natural tangent is 21° 48'. 

The vertical tension at each point 

of suspension is = half the weight = 

250000 ; the horizontal tension at the 

points of suspension = 250000 X cot. 

250000 
21° 48' = .^ = 625000 lbs. 

The whole tension at one end will be 



x/6250002 + 250000^ = 673146 lbs. 



4. Suppose the piston of a steam 
engine, with its rod, weighs 1000 lbs. ; 
it has no velocity at its highest and 
lowest positions, but in the middle the 
velocity is a maximum and equal 10 ft. ; 
what effect will it accumulate by virtue 
of its inertia in the first half of its 
path, and give out again in the second 
half ; and what is the mean force which 
would be requisite to accelerate the 
motion of the piston in the first half 
of its path, which is the same as that 
which it would exert in the second half 
by its retardation, the length of stroke 
being 8 feet. 

According to the principle of vis 
viva, the effect which the piston will 
accumulate by virtue of its inertia in. 
the first half of its path, and give 
out again in the second half = 

10^ 
^ ^ g2-2 ^ -^^^^ "" 1552-794 units of work. Half the path of 

the piston = 4 feet ; hence, 
1552-794 
J = 388-1985 lbs., the mean force. 

Moment of Inertia, or the Moment of Rotation, or the 
Moment of the Mass, is the sum of the products of the particles 



MOMENT OF INERTIA. 



415 



of the mass and the squares of their distances from the axis of 
rotation. 

5. If a body at rest, but capable of turning round a fixed axis A, 
possesses a moment of inertia of 121 units of work, the measures 
taken in feet and pounds, made to turn by means of a cord and 
weight of 36 lbs., lying over a pulley in a path of 10 feet; what 
are the circumstances of the motion. 



4 



2 X 36 X 10 



— — j = 2-439347 feet, the angular velocity of the 

body, which call v ; so that each point at the distance of one foot 
from the axis of revolution will describe, after the accumulatior 
of 121 units of work, 2-44 feet in a second. 

6*2832 = circumference of a circle 2 feet in diameter, 

6-2832 ^ ^ 
Q,. . = 2-6 seconds, the time of one revolution. 

. 6. If an angular velocity of 3 feet passes into a velocity of 7 feet ; 
what mechanical effect will a mass produce so moving, supposing 
the moment of inertia to be 200, the measures taken in feet and 
pounds. 

According to the principles of vis viva, 

200 
(7^ - 3^) -2- = 4000 units of work, which may be 40 lbs. 

raised 100 feet, 80 lbs. raised 50 feet, 400 lbs. raised 10 feet ; and 
so on. 

7. The weight of a 
rotating mass B is 
500 lbs., its distance 
OB from the axis of 
rotation 3 feet, the 
weight W, constitut- 
ing the moving force, 
90 lbs., its arm AO 
= 00 = 4 feet ; re- 
quired the circum- 
stances of the motion 
that ensues, 

I 90 + - 500 I -T- 
\ 42 / 

32-2 = 11-53 lbs., 
the inert mass accele- 
rated by the force of 
W. And it is well 
known that the force 
divided by the mass 
gives the accelera- 
tion. 




416 



THE PEACTICAL MODEL CALCULATOR. 



90 

.'. -i-,ro = T*806, the acceleration of the motion of W. The 

... . . T-806 

angular acceleration in a circle 1 foot from the axis = — r— = 

1-9515. ^ 

After 10 seconds the acquired angular velocity will be 

1-9515 X 10 = 19-515. 

1*9515 X 10^ 
And the corresponding distance = ^ ~ 97-575 feet, 

measured on a circle one foot from 0. 



The space described by the weight W is 



7-806 X 10^ 



= 390-3 feet, 



which is the same as the space described by C. The circumfe- 
rence of a circle one foot from C = 3-1416. 

97-575 
.*. Q.-| .-.^ = 31-059 revolutions. 

In the rotation of a body AB about a fixed axis 0, all its points 
describe equal angles in equal times. If the body rotate in a cer- 

tain time through the angle e°, or arc 9 ~Tso°'*' ^^^^^^ ~ -'- > 

and hence, Jt = 3*141592, &c. ; the elements of the body, a, b, c, &c., 
at the distances 



oa 
axis 


= x^, oh = 
&c. from the 
, will describe 


the 
a a. 


arcs or spaces 
= ^x^, h\ = 




&c. If the angu- 
lar velocity, that 
is, the velocity of 
those points of 
the body which 
are distant a unit 
of length, a foot, 
from the axis of 
revolution, be put 
= z, then the si- 
multaneous velo- 
cities of the ele- 
ments of the mass at the distances x^, x^, x^, &c., will be, 

zx^, zx^, zx^, &c. 
And if a be the mass of the element at a ; h the mass of the ele- 
ment at b ; c the mass of the element at c, &c., their vis viva will be, 

{zx.Ya, [zx.fb, {zxje, kc. 
And the sum of the vis viva of the whole body = 

z^ {x^ a + a:/5 + x^ c^ &c.) 



MOMENT OF INERTIA. 417 

According to our definition, x^ a + x^h + x^c^ kc. is the mo- 
ment of inertia, which may be represented by R ; then 2^R is the 
vis viva of a body revolving with the angular velocity z. There- 
fore, to communicate to a body in a state of rest an angular velo- 
city 2, a mechanical effect F s, or force X space = f the vis viva, 
must be expended ; that is, F s = J 2;^ R, or, which is the same thing, 
a body performing the units of work F s, passes from the angular 
velocity 2 to a state of rest. In general, if the initial angular 
velocity = v, and the terminal angular velocity = z, the units of 
work will be, 

z^ — v"- 
F« = — 2 — ^ ^• 

The moment of inertia of a body about an axis not passing 
through the centre of gravity is equivalent to its moment of inertia 
about an axis running parallel to it through the centre of gravity, 
increased by the product of the mass of the body and the square of 
the distance of the two centres. 

It is necessary to know the moments of inertia of the principal 
geometrical bodies, because they very often come into application 
in mechanical investigations. If these bodies be homogeneous, as 
in the following we will always suppose to be the case, the particles 
of the mass M^, M^, &c. are proportional to the corresponding par- 
ticles of the volume V^, V^, &c. ; and hence the measure of the 
moment of inertia may be replaced by the sum of the particles of 
the volume, and the squares of their distances from the axis of 
revolution. In this sense, the moments of inertia of lines and sur- 
faces may also be found. 

If the whole mass of a body be supposed to be collected into one 
point, its distance from the axis may be determined on the suppo- 
sition that the mass so concentrated possesses the same moment 
of inertia as if distributed over its space. This distance is called 
the radius of gyration, or of inertia. If R be the moment of iner- 
tia, M the mass, and r the radius of gyration, we then have M r^ = 

/R 

R, and hence ^ = \ a?- ^^ Timsi bear in mind that this radius by 

no means gives a determinate point, but a circle only, within whose 
circumference the mass may be considered as arbitrarily distributed. 

If into the formula R^ = R + M e^, expressed in the words above 
printed in italics, we introduce R = Mr" and R^ = M r^, we 
obtain r^ — 7^ -\- e^', that is, the square of the radius of gyration 
referred to a given axis = the square of the radius of gyration 
referred to a parallel line of gravity, plus the square of the dis- 
tance between the two axes. 

Wheel and axle. — The theory of the moment of inertia finds ita 
most frequent application in machines and instruments, because in 
these rotary motions about a fixed axis are those which generally 
present themselves. 

27 



418 



THE PRACTICAL MODEL CALCULATOR. 



If two weights, P and Q, act on a wheel and axle ACDB, with 
the arms CA = a and DB = h through the medium of perfectly 
flexible strings, and if the radius of the gudgeons be so small that 
their friction may be neglected, it will remain in equilibrium if the 
statical moments P . CA and Q . DB are equal, and therefore 
P « = Q 5. But if the moment of the weight P is greater than 
that of Q, therefore P « > Q 5, P will descend and Q ascend ; if 
P a < Q 5, P will ascend and Q descend. Let us now examine the 




conditions of motion in the case that P a > Q 5. The force corre- 
sponding to the weight Q and acting at the arm h generates at the 

Q5 . . 

arm a a force — , which acts opposite to the force corresponding 

to the weight P, and hence there is a residuary moving force 
P acting at A. The mass — is reduced by its transference 

from the distance b to that of a to — r ; hence the mass moved by 



Qh 



Qh 



P — — is M = (^P H ^) -T- g^ or, if the moment of inertia of 



MOMENT OF INERTIA. 419 

the wheel and axle without the weights P and Q = , and, there- 

fore, its inert mass reduced to A = — ^, we have, more exactly, 

M = (P + ^ + ^) ^ 5. = (Pa' + Q J« + Qf)-^ga\ 

From thence it follows that the accelerated motion of the weight 
P, together with that of the circumference of the wheel, namely, 

moving force a 2_ P«--Q6 

P~ ^ais^ -pV+"QFfGy^'* ^¥¥+W+W^^' 
on the other hand, the accelerated motion of the ascending weight 
Q, or of the circumference of the axle, is, 
_b_ __ Pa- Q5 
^~a^ ~Pa^+ Q62 + G?/^^^* 

The tension of the string by P is S = P =^ = P (l - -), 

Qg / q\ 

that of the string by Q is T = Q H = Q (^1 + -j; hence the 

pressure on the gudgeon is, 

g + T-P + Q-g^ + ^-P + Q- (P^-Q^)^ . 

the pressure, therefore, on the gudgeons for a revolving wheel and 
axle is less than for one in a state of equilibrium. Lastly, from 
the accelerating forces p and q, the rest of the relations of motion 
may be found ; after t seconds, the velocity of P is v = pt, of Q 
is v^ == q t, and the space described by P is s = ^ 2^^% hy Q is 
S, = hqt\ 

Let the weight P at the wheel be = 60 lbs., that at the axle 
Q = 160 lbs., the arm of the first pA = <z = 20 inches, that of 
the second DB = b = 6 inches ; further, let the axle consist of a 
solid cylinder of 10 lbs. weight, and the wheel of two iron rings 
and four arms, the rings of 40 and 12 lbs., the arms together of 
15 lbs. weight; lastly, let the radii of the greater ring AE == 
20 and 19 inches, that of the less FG = 8 and 6 inches; required 
the. conditions of motion of this machine. The moving force at 
the circumference of the wheel is, , 

p _ ^ Q = 60 - ^ 160 = 60 - 48 = 12 lbs., 

the moment of inertia of the machine, neglecting the masses of the 
gudgeons and the strings, is equivalent to the moment of inertia 

W b^ 10 , 6^ 

of the axle = — o~ = — n — = 180, plus the moment of the smaller 

1^1 (^' + rJ") 12 (8^ + 62) ^^^ , , 
rmg = 2 — - — ■ ~ ~ 600, plus the moment of 



420 THE PRACTICAL MODEL CALCULATOR. 

40 (20^ + 192) 
the larger ring = ^ — 15220, plus the moment of 

,, • .1 A (p,^ - p/) A p/ + P,P, + P/) 
the arms, approximately = -w^ — ^^ — r- = q = 

15 fl9^ 4- 19 X 8 4- S^'i 

— ^ g ^ = 2885 ; hence, collectively, G-^^ = 180 + 

18885 
600 + 15220 + 2885 = 18885, or for foot measure = ^^g- = 

131 '14. The collective mass, reduced to the circumference of the 

■wheel is, 

/^ W-\-(^y\ r.r. ... /6\' 18885^ 

= (P + ^^^^i-^h^=[60 + 160(2o) +-20^]^^ = 

(60 + 160 X 0-09 + '^^) 0-031 = 121-61 x 0-031 = 337 lbs. 

Accordingly, the accelerated motion of the weight P, together 
with that of the circumference of the wheel, is. 



^-^Q 12 



h 
P "" p-^ Q52^ Q^2 9 = ^ ■ = 3-183 feet; on the other 

h 6 

hand, that of Q is ^^ = - 1? = ^q 3*183 = 0-954 feet ; further, the 

/ »\ / 3-133x 

tension of the string by P is = (^1 — -J P = (^1 — -^^) 60 = 



g^ " V" 32-2 



54-07 lbs. ; that by Q, on the other hand, Q = (l + -) Q = 

(1 + 0-925 X 0-032) 160 = 1-030 160 = 164-8 lbs. ; and con- 
sequently the pressure on the gudgeons S + T = 54-06 -f 164-80 = 
218-86 lbs., or inclusive of the weight of the machine = 218-86 + 
77 = 295-86 lbs. After 10 seconds, P has acquired the velocity 
pt = 3-084 X 10 = 30-84 feet, and described the space s = 

— = 30-84 X 5 = 154-2 feet, and Q has ascended a height - s = 

0-3 X 154-2 = 46-26 feet. 

The weight P which communicates to the weight Q the accele- 
^ . Vah-Qh^ . . ... 

rated motion q = p ^2 ■ q ^2 . (j. ^2 9^ ^^1 also be replaced by 

another weight P^, without changing the acceleration of the motion 
Q, if it act at the arm a^, for which, 

P,g, - Q5 Va - Qb 

P,< + Qb' + Gf ~ Pa^ + Q^ + Grf 

, Pa2 + Q¥ + Gf 
The magnitude V a — Ob ' ''^P^'^sented by Jc, and we ob- 

. ^ ^ . Qbjb + ^) + Gf , ^ 
tain ftj , — ka^ — — p , and the arm in question, 



MOMENT OF INERTIA. 421 



a, = ik±: yj{-^) p . 

We may also find by help of the differential calculus, that the mo- 
tion of Q is most accelerated by the weight P, when the arm of the 
latter corresponds to the equation Va^ — 2Qab = Qb^ + G ^/^ 
therefore, 

b Q //6Qx2 + Qb' + Gf 
^ = T" + JvT") P • 

The formula found above assumes a complicated form if the fric- 
tion of the gudgeons and the rigidity of the cord are taken into 
account. If we represent the statical moments of both resistances 

by F r, we must then substitute for the moving force P Q, the 

, ^ Q5 + Fr 
value P , whence the acceleration of Q comes out, 

(Pa-Fr)b-Qb' ^ Qb+Fr hQb+¥r^'+Qb'+Gy' 
^^p^ + Q-jq:^?^ and a =— p— + J(^ '-T~~) P * 

The weights P = 30 lbs. Q = 80 lbs. act at the arms a = 2 feet, 
and 5 = 1 foot of a wheel and axle, and their moments of inertia 
Gy^ amount to 60 lbs. ; then the accelerated motion of the ascend- 
ing weight Q is, 

_ 30 X 2 X I - 80 X {JY _ 30 - 20 322 

^~30x22 + 8qx(i)2+60^ ~ 120 + 20 + 60 ^^'^ ~ 200 ~ 
1*61 feet. But if a weight P^ = 45 lbs. generates the same acce- 
leration in the motion of Q, the arm of P^ is then. 



^ /.y__ 8QxH^ + ^) + 60 200 

2 n/V2/ 45 ? or as a; - gQ _ ^^ 



J 32 

25 - y = 5 db J 11-358 = 5 +: 3-786 = 8-786 

feet, or 1-214 feet. 

The accelerated motion of Q comes out greatest if the arm of 
the force or radius of the wheel amount to. 



1x80 //40n2 20 + 60 4 116 24 4 + v/40 



//40n2 20 + 60 4 I 



"*"" 30 "^JV307 "^ 30 ~3"^n19"^9"~ 3 ~ 
. ...r o . - /30 X 1-7207 - 20x 31-621 

3-4415 f6et, and q is = (30 x (3-4415)^ + 80 ) ^ = 435^ ^ = 
2-339 feet. 

The statical moment of the friction, together with the rigidity 
of the string, is F r = 8 ; then, instead of Q b, we must put Qb + 
Fr = 40 + 8 = 48 ; w hence it follows that, 

a = g^ + J{^) + g = 1-6 + x/5^227 = 3-886, and the cor- 
respondent maximum accelerating force 

30 X 1-943 - 8 X J - 20 34-29 ^^ ^ o n^i ^ . 

? = 30 X (3-886)-+ 80 ^ = 53^ ^ ^^'^ = ^'^^^ ^'''' 



422 



WEIGHT, ACCELERATION, AND MASS. 

PARALLELOGRAM OF FORCES. — ^THE PRINCIPLE OP VIRTUAL VELOCITIES. 

MECHANICAL POWERS: CONTINUOUS CIRCULAR MOTION, GEARING, 

TEETH OF WHEELS, DRUMS, PULLEYS, PUMPING ENGINES, ETC. 

1. If a weight of 10 lbs., moved by the hand, ascends with a 3 
feet acceleration, what is the pressure on the hand ? 

10 (1 -f 3I2) = 10-93168 lbs. 

If a weight of 10 lbs., moved by the hand, descends with a 3 feet 
acceleration, the pressure on the hand will be 9-06832 lbs., for then 

10 (1 - 32.2) = 9-06832. 

If w be the weight of the mass acted upon by the force of the 

hand, and also by the force of gravity, as ^ = 32-2, ,the mass 

w 
moved by the sum or difference of these forces will be = — . If P 

be the pressure on the hand, and p its acceleration, the body falls 

•* w 
with the force — ^ ; it also falls with the force w — P ; hence, 

W V 

if y 

When the body is ascending, then p is negative, 

w V 

and ^(; + P = - (- p) .*. P = (1 + -) «^. 

y y 

2. If a body of 200 lbs. be moved on a smooth horizontal track, 
by the joint action of two forces, and describes a space of 10 feet 
in the first second, what is the amount of each of these forces ; the 
first makes an angle of 35° with the track upon which the body 
moves, and the other an angle of 50° ? 

In solving this question, the natural sines of the angles 35°, 50°, 
and of their sum 85°, will be required. We shall first take these 
from the table : 

sin. 35° = -57358 

sin. 50° = -76604 

sin. 85° = -99619. 
The acceleration is == 20 feet, that is, twice the space passed over 
in the first second, 

200 200 

^^ = the mass, and-g^:^ X 20 = 124*224 lbs., the force of 

the resultant, in the direction of the track upon which the body 
moves. 



WEIGHT, ACCELERATION, AND MASS. 



423 



One of the components 



sin. (35° + 50°) 



= 71-52 lbs. 



124-224 sin. 50° ^. .^ „ 
The other component = -; — /q^q ■ 5qo\ = 95*52 lbs. 

These, and the like results, may be obtained with greater ease 
by logarithms. 



Log. 124-224 = 

Log. sin. 35° = 

Log. sin. 85° = 

Log. of 71-52413 = 

Log. 124-224 = 

Log. sin. 50° = 

Log. sin. (85°) = 

Log. of 95-5247 = 



2-0942055 
9-7585913 

11-8527968 
9-9983442 

1-8544526 

2-0942055 
9-8842540 

11-9784595 
9-9983442 



1-9801153 

3. A carriage weighing 8000 lbs. is moved forward by a force /^ 
of 500 lbs. upon a horizontal surface AB ; during the motion, two 
resistances have to be overcome, one horizontal of 100 lbs., the 
amount of friction, represented in the figure by/3, the other /g of 




200 lbs. acting downwards ; the angles f^ nf^ and / n m, which the 
directions of these forces make with the horizon, are 61° and 21° 
respectively : it is required to know what work the force /^ will 
perform by converting a 5 feet initial velocity of the carriage into 
a 20 feet velocity. 

If we put X = nm, the distance the carriage moves in passing 
from a 5 to a 20 feet velocity. 

The work of the force/ =f^xnq = 500 X cos. 21° x x. 

The work of the force/ = (— /) X 5im = — 100 X x. 

•The work of the force/; = (-/,) xnp = -200x cos. 61° X x. 



424 



THE PRACTICAL MODEL CALCULATOR. 



Consequently, the work of the effective force will be 269-828 x 
X = {500 X -94358 - 100 - 200 x -48481} x, since the natu- 
ral cosine of 21° = -93358, and the natural cosine of 61° = -48481. 
But according to the principle of vis viva^ the work done is 
equal to 

20^ — 52 

X 8000 = 46589-82. 



64-4 
.-. 269-828 xx = 46589-82 and x = 



46589-82 



269-828 
772-665 feet, the space passed over by the carriage. 

This question is solved on the principle of virtual velocities, 
which we shall explain, as it is of essential service in practical 
mechanics. 

This explanation depends on what is technically termed the 
" Parallelogram of Forces.'' 




When a material point 0, is acted upon by two forces /i,/^, whose 
directions 0/^, O/3, make with each other an angle, if O/j, O/3 re- 
present the magnitudes and directions of the forces, the diagonal 
of the parallelogram /^ f^ f^ represents the resultant in magnitude 
and direction ; that is, the diagonal represents a single force equal 
to the combined actions of the forces represented by the sides. 
And if the sides of the parallelogram represent the accelerations 
of the forces, the diagonal represents the resultant acceleration. 
Draw through 0, two axes OX and OY, at right angles to each 
other, and resolve the forces /^ and/^, as well as their resultant /g, 
into components in the directions of these axes ; namely, /^ into n^ 
and m^ ; /^ into n^ and m^ ; and f^ into n^ and m„. The forces in 
one axis are w^, n^^ and n^ ; and those in the other 7??^, m^, and w,. 
And by the parallelogram of forces it is well known that 

^3 — '^^1 + '^2 ^^^ ^^3 = '^^1 + '^^3- (^)- 
^Now if we take in the axis OX any point P, and let fall from'it 



MECHANICAL POWERS. 425 

the perpendiculars PA, PB, PC, on the directions of the forces /^, 

/g,/^, we obtain the following similar right-angled triangles, namely, 

OAP and n^f^ are similar ; 

OBP and nj^ ; 

OOP and 71,/, ; 

On, OA n, ^ AO ^ ^ . ., , ^ 

.*. Qj = Qp = -^ and n, = 'TTp/i* -l-t is easily seen also that 

„=C0 iBO 

• If the values be substituted in (E), we obtain 
BO x/3^C0 x/,+ AO x/,. 
From the similarity of these triangles, and the remaining equa- 
tion of (E), we can readily find that 

PB x/3=PA x/,+ PC x/,. 
The equation becomes more compact by putting 

OA, 00, OB, respectively equal s,, s^ s^ ; and 

PA, PC, PB, g„ q^, q^. 

Then/353 =/,s, + /,s, ^ridf^q^=f„^q^ + f^q,. 
The same holds good with any number of forces /j,/j,/3, &c., 
and their resultant /„, that is 

and/„g„ = /,^, + /,g, + /3g3 + &c. 
If the point of application 0, move in a straight line to P, then 
OA = s, is called the space of the force/, and/^ s^ the work done 
by the force /, in moving the body from to P. OB is the space 
of the resultant, and the product /3S3, the work done by it. f^s^ 
is the work done by/, in moving the material point from to P. 
Hence the work done by the resultant is equal to all the work done 
by the component forces, as we have shown. 



PRINCIPLES AND PRACTICAL APPLICATIONS OF 
MECHANICAL POWERS. 

Mechanical Powers, or the Elements of Machinery, are certain 
simple mechanical arrangements whereby weights may be raised or 
resistances overcome with the exertion of less power or strength 
than is necessary without them. 

They are usually accounted six in number, viz. the lever, the 
tvJieel and axle, the pulley, the incliiied plane, the wedge, and the 
screw ; but properly two of these comprise the whole, namely, the 
lever and inclined plane, — the wheel and axle being only a lever 
of the first kind, and the pulley a lever of the second, — the wedge 
and the screw being also similarly allied to that of the inclined 
plane : however, although such seems to be the case in these re- 



426 



THE PRACTICAL MODEL CALCULATOR. 



spects, yet they each require, on account of their various modifica- 
tions, a peculiar rule of estimation adapted expressly to the difter- 
ent circumstances in which they are individually required to act. 



THE LEVER. 



Levers, accDrding to mode of application, as the following, are 



1st. 



2nd. 



3rd. 



7 lbs. 



56 lbs. 



8 lbs. 




B A 



56 lbs. 



distinguished as be 
ing of the first, se- 
cond, or third kind ; 
and although levers p^ 
ofequal lengths pro- ^"Gl 
duce different ef- ^ 
fects, the general 
principles of esti- 
mation in all are 
the same ; namely, 
the power is to the 
weight or resistance, as the distance of the one end to the fulcrum 
is to the distance of the other end to the same point. 

In the first kind, the power is to the resistance, as the distance 
AB is to the distance BC. 

In the second, the power is to the resistance, as the distance AB 
is to that of AC ; and. 

In the third, the resistance is to the power, as the distance AB 
is to that of AC. 

Rule, first kind, — Divide the longer by the shorter end of the 
lever from the fulcrum, and the quotient is the effective force that 
the power applied is equal to. 

Let the handle of a pump equal 65 inches in length, and 10 
inches from the shortest end to centre of motion; what is the 
amount of effective leverage thereby obtained ? 

55 
65 ~ 10 = 55, and ta = 5} to 1. 

Required the situation of the fulcrum on which to rest a lever 
of 15 feet, so that 2J cwt. placed at one end may equipoise 30 cwt. 
at the other, the weight of the lever not being taken into account. 

15 X 2*5 

^r , QQ = 1-154 feet from the end on which the 30 cwt. is to 

be placed. 

It is by the second kind of lever that the greatest effect is ob- 
tained from any given amount of power; hence the propriety of 
the application of this principle to the working of force pumps, and 
shearing of iron, as by the lever of a punching-press, &c. 

'RjJL^, second kind. — Divide the whole length of lever, or dis- 
tance from power to fulcrum, by the distance from fulcrum to 
weight, and the quotient is the proportion of effect that the power 
is to the weight or resistance to be overcome. 

Required the amount of effect or force produced by a power of 



MECHANICAL POWERS. 427 

50 lbs. on the ram of a Bramah's pump, the length of the lever 
being 3 feet, and distance from ram to fulcrum 4J inches. 

3 feet = 36 inches, and r^ = 8, or the power and resistance 

are to each other as 8 to 1 ; hence 50 x 8 = 400 lbs. force upon 
the ram. 

The lever on the safety valve of a steam boiler is of the third 
kind, the action of the steam being the power, and the weight or 
spring-balance attached the resistance ; but in such application the 
action of the lever's weight must also he taken into account. 

THE WHEEL AND PINION, OR CRANE. 

The mechanical advantage of the wheel and axle, or crane, is as 
the velocity of the weight to the velocity of the power ; and being 
only a modification of the first kind of lever, it of course partakes 
of the same principles. 

Rule. — To determine the amount of effective power produced 
from a given power hy means of a crane with known peculiarities. — 
Multiply together the diameter of the circle described by the winch, 
or handle, and the number of revolutions of the pinion- to 1 of the 
wheel ; divide the product by the barrel's diameter in equal terms 
of dimensions, and the quotient is the efi'ective power to 1 of ex- 
ertive force. 

Let there be a crane the winch of which describes a circle of 30 
inches in diameter ; the pinion makes 8 revolutions for 1 of the 
wheel, and the barrel is 11 inches in diameter ; required the effec- 
tive power in principle, also the weight that 36 lbs. would raise, 
friction not being taken into account. 

30 X 8 

— ^j— = 21-8 to 1 of txertive force ; and 21-8 x 36 = 784-8 lbs. 

Rule. — G-iven any two parts of a crane, to find the third, that 
shall produce any required 'proportion of mechanical effect. — Mul- 
tiply the two given parts together, and divide the product by the 
required proportion of effect ; the quotient is the dimensions of the 
other parts in equal terms of unity. 

Suppose that a crane is required, the ratio of power to effect 
being as 40 to 1, and that a wheel and pinion 11 to 1 is unavoid- 
ably compelled to be employed, also the throw of each handle to 
be 16 inches ; what must be the barrel's diameter on which the 
rope or chain must coil ? 

16 X 2 = 32 inches diameter described by the handle. 

32 X 11 
And — -TK — = 8*8 inches, the barrel's diameter. 

THE pulley. 
The principle of the pulley, or, more practically, the block and 
tackle, is the distribution of weight on various points of support ; 
the mechanical advantage derived depending entirely upon the 



428 THE PRACTICAL MODEL CALCULATOR. 

flexibility and tension of the rope, and the number of pulleys or 
sheives in the lower or rising block : hence, by blocks and tackle of 
the usual kind, the power is to the weight as the number of cords 
attached to the lower block ; whence the following rules. 

Divide the weight to be raised by the number of cords leading 
to^ from, or attached to the lower block ; and the quotient is the 
power required to produce an equilibrium, provided friction did not 
exist. 

Divide the weight to be raised by the power to be applied ; the 
quotient is the number of sheives in, or cords attached to the rising 
block. 

Required the power necessary to raise a weight of 3000 lbs. by 
a four and five-sheived block and tackle, the four being the mov- 
able or rising block. 

Necessarily there are nine cords leading to and from the rising 

block. 

3000 
Consequently — q— = 333 lbs., the power required. 

I require to raise a weight of 1 ton 18 cwt., or 4256 lbs. ; the 
amount of my power to effect this object being 500 lbs., what kind 
of block and tackle must I of necessity employ ? 

4256 

-^QQ- = 8-51 cords ; of necessity there must be 4 sheives or 9 

cords in the rising block. 

As the effective power of the crane may, by additional wheels 
and pinions, be increased to any required extent, so may the pulley 
and tackle be similarly augmented by purchase upon purchase. 

THE INCLINED PLANE.. 

The inclined plane is properly the second elementary power, and 
may be defined the lifting of a load by regular instalments. In 
principle it consists of any right line not coinciding with, but ly- 
ing in a sloping direction to, that of the horizon ; the standard of 
comparison of which commonly consists in referring the rise to so 
many parts in a certain length or distance, as 1 in 100, 1 in 200, 
&c., — the first number representing the perpendicular height, and 
the latter the horizontal length in attaining such height, both num- 
bers being of the same denomination, unless otherwise expressed ; 
but it may be necessary to remark, that the inclination of a plane, 
the sine of inclination, the height per mile, or the height for any 
length, the ratio, &c., are all synonymous terms. 

The advantage gained by the inclined plane, when the power acts 
in a parallel direction to the plane, is as the length to the height 
or angle of inclination : hence the rule. Divide the weight by the 
ratio of inclination, and the quotient equal the power that will just 
support that weight upon the plane. Or, multiply the weight by 
the height Of the plane, and divide by the length, — the quotient is 
the power. 



MECHANICAL POWERS. 



429 



Required the power or equivalent weight capable of supporting 
a load of 350 lbs. upon a plane of 1 in 12, or 3 feet in height and 
36 feet in length. 

350 X 3 
or — 7^ — = 29-16 lbs. power, as before. 



S50 

-^ = 29-16 lbs., 



36 



The weight multiplied by the length of the base, and the product 
divided by the length of the incline, the quotient equal the pres- 
sure or downward weight upon the incline. 



Table showing the Resistance opposed to the Motion of Caro 
on different Inclinations of Ascending or Descending Planes^ 
whatever part of the insistent weight they are drawn hy. 



a 


Hundreds. 




100 


200 


300 


400 


500 


600 


700 
•00143 


800 


900 .! 






•01 


•005 


•00333 


•0025 


•002 


•00167 


•00125 


•00111 


10 


•1 


•00909 


•00476 


•00322 


•00244 


•00196 


•00164 


•00141 


•00123 


•0011 


20 


•05 


•00833 


■00454 


•00312 


•00238 


•00192 


•00161 


•00139 


•00122 


•00109 


30 


•0333 


•00769 


•00435 


•00303 


•00232 


•00189 


•00159 


•00137 


•0012 


•00107 


40 


•025 


•00714 


•00417 


•00294 


•00227 


•00185 


•00156 


•00135 


•00119 


•00106 


50 


•02 


•00667 


•004 


•00286 


•00222 


•00182 


•00154 


•00133 


•00118 


•00105 


60 


•0166 


•00625 


•00385 


•00278 


•00217 


•00178 


•00151 


•00131 


•00116 


•00104 


70 


•0143 


•00588 


•0037 


•0027 


•00213 


•00175 


•00149 


•0013 


•00115 


•00103 


80 


•0125 


•00555 


•00357 


•00263 


•00208 


•00172 


•00147 


•00128 


•00114 


•00102 


90 


•0111 


•00526 


•00345 


•00256 


•00204 


•00169 


•00145 


•00126 


•00112 


•00101 



Although this table has been calculated particularly for car- 
riages on railway inclines, it may with equal propriety be applied 
to any other incline, the amount of traction on a level being known. 

Application of the preceding Table. 
What weight will a tractive power of 150 lbs. draw up an incline 
of 1 in 340, the resistance on the level being estimated at ^th 
part of the insistent weight ? 

In a line with 40 in the left-hand column and under 200 is -00417 
Also in the same line and under 390 is -00294 

Added together = -00711 

150 
Then ^^^^-.-j = 21097 lbs. weight drawn up the plane. 

What weight would a force of 150 lbs. draw down the same plane, 
the fraction on the level being the same as before ? 
Friction on the level = -00417 
Gravity of the plane = -00294 subtract 

= -00123 
150 
And TqaToo = 121915 lbs. weight drawn down the plane. 

Example of incline when velocity is taken into account. — A power 
of 230 lbs., at a velocity of 75 feet per minute, is to be employed 
for moving weights up an inclined plane 12 feet in height and 163 



430 



THE PRACTICAL MODEL CALCULATOR. 



feet in length, the least velocity of the weight to be 8 feet per 
minute ; required the greatest weight that the power is equal to. 

230 X 75 X 163 2811T50 

= 29288 lbs., or 13-25 tons. 



12 X 



96 



Table of Inclined Planes, showing the ascent or descent per yard, 
and the corresponding ascent or descent per chain, per mile; and 
also the ratio. 



Per 


yard. 


Per chain. 


Per mile. 


Ratio. 


Pel 


■ yard. 


Per chain. 


Per mile. 


Ratio. 


In parts 
of an in. 


In dec-Is. 
jf an inch. 


Inches. 


Feet. 


1 inch. 


In parts 
of an in. 


In decimals 
of an inch. 


Inches. 


Feet. 


linch. 


^ 


•0156 


•344 


2-29 


2304 


tV 


•4375 


9-625 


64-17 


82 


^H 


•0208 


•458 


3-06 


1728 


\ 


•5 


11 


73-33 


72 


^ 


•0312 


•687 


4-58 


1152 


1% 


•5625 


12-375 


82-5 


64 


^ 


•0417 


•917 


6-11 


864 


tV 


•5833 


12-833 


85-56 


62 


' A 


•0625 


1^375 


9^17 


576 


1 


•6 


13-2 


88 


60 


tV 


•0833 


1^833 


12-22 


432 




•625 


13-75 


91-67 


58 


A 


•1 


2-2 


14-67 


360 


f 


•6667 


14-667 


97-78 


54 


* 


•125 


2-75 


18-33 


288 


H 


•6875 


15-125 


100-83 


52 


i 


•1667 


3^667 


24-44 


216 


A 


•7 


15-4 


102-67 


51 


^ 


•1875 


4-125 


27-50 


192 


f 


•75 


16-5 


110 


48 


i 


•2 


4-4 


29-33 


180 


f 


•8 


17-6 


117-33 


45 


i 


•25 


5^5 


36-67 


144 


If 


•8125 


17-875 


119-17 


44 


A 


•3 


6^6 


44 


120 


^ 


•8333 


18-333 


122-22 


43 


A 


•3125 


6-875 


45-83 


115 


i 


•875 


19-25 


128-33 


41 




•3333 


7^333 


48-89 


108 


A 


•9 


19-8 


132 


40 


f 


•375 


8-25 


55 


96 


fi 


•9167 


20-167 


134-44 


39 


f 


•4 


8-8 


58-67 


20 


if 


•9375 


20-625 


137-5 


38 


A 


•4167 


9^167 


61-11 


•86 


1 


1 


22 


146-67 


36 



THE WEDGE. 

The wedge is a double inclined plane ; consequently its principles 
are the same : hence, when two bodies are forced asunder by means 
of the wedge in a direction parallel to its head, — Multiply the re- 
sisting power by half the thickness of the head or back of the wedge, 
and divide the product by the length of one of its inclined sides ; 
the quotient is the force equal to the resistance. 

The breadth of the back or head of a wedge being 3 inches, and 
its inclined sides each 10 inches, required the power necessary to 
act upon the wedge so as to separate two substances whose resist- 
ing force is equal to 150 lbs. 

150 X 1-5 ^ 

^ = 22-5 lbs. 

When only one of the bodies is movable, the whole breadth of 
the wedge is taken for the multiplier. 

THE SCREW. 

The screw, in principle, is that of an inclined plane wound around 
a cylinder, which generates a spiral of uniform inclination, each 
revolution producing a rise or traverse motion equal to the pitch 
of the screw, or distance between two consecutive threads, — the 
pitch being the height or angle of inclination, and the circumference 



( 



MECHANICAL POWERS. 431 

the length of the plane when a lever is not applied ; but the' lever 
being a necessary qualification of the screw, the circle which it de- 
scribes is taken, instead of the screw's circumference, as the length 
of the plane : hence the mechanical advantage is, as the circum- 
ference of the circle described by the lever where the power acts, 
is to the pitch of the screw, so is the force to the resistance in 
principle. 

Required the effective power obtained by a screw of J inch pitch, 
and moved by a force equal to 50 lbs. at the extremity of a lever 
30 inches in length. 

30x2x3-1416x60 ,,^,, „ 
:gYg = 10760 lbs. 

Required the power necessary to overcome a resistance equal to 
7000 lbs. by a screw of 1^ inch pitch, and moved by a lever 25 
inches in length. 

7000 X 1-25 
25 X 2 X 3-1416 = ^^'^^ ^^'- P^^^^* 

In the case of a screw acting on the periphery of a toothed wheel, 
the power is to the resistance, as the product of the circle's circum- 
ference described by the winch or lever, and radius of the wheel, 
to the product of the screw's pitch, and radius of the axle, or point 
whence the power is transmitted ; but observe, that if the screw 
consist of more than one helix or thread, the apparent pitch must 
be increased 'so many times as there are threads in the screw. 
Hence, to find what weight a given power will equipoise : 

Rule. — Multiply together the radius of the wheel, the length of 
the lever at which the power acts, the magnitude of the power, and 
the constant number 6*2832 ; divide the product by the radius of 
the axle into the pitch of the screw, and the quotient is the weight 
that the power is equal to. 

What weight will be sustained in equilibrio by a power of 100 
lbs. acting at the end of a lever 24 inches in length, the radius of 
the axle, or point whence the power is transmitted, being 8 inches, 
the radius of the wheel 14 inches, the screw consisting of a double 
thread, and the apparent pitch equal f of an inch ? 

14x24x100x6-2832 ^,,,, ,. „ 

^25 X 2 X 8 ~ 21111-5o lbs., or 9-4 tons, the 

power sustained. 

If an endless screw be turned by a handle of 20 inches, the threads 
, of the screw being distant half an inch ; the screw turns a toothed 
wheel, the pinion of which turns another wheel, and the pinion of 
this another wheel, to the barrel of which a weight W is attached ; 
it is required to find the weight a man will be able to sustain, who 
acts at the handle with a force of 150 lbs., the diameters of the 
wheels being 18 inches, and those of the pinions and barrel 2 inches. 
150 X 20 X 3-1416 x 2 x 18^ = W x 2^ x i ; 

.-. W = 12269 tons. \ 



432 THE PRACTICAL MODEL CALCULATOR. 



CONTIinJOUS CIRCULAR MOTION. 

In mecliaiiics, circular motion is transmitted by means of ivheels, 
drums, or pulleys ; and accordingly as the driving and driven are 
of equal or unequal diameters, so are equal or unequal velocities 
produced : hence the principle on which the following rules are 
founded. 

KuLE. — When time is not taken into account. — Divide the greater 
diameter, or number of teeth, by the lesser diameter, or number 
of teeth, and the quotient is the number of revolutions the lesser 
will make for 1 of the greater. 

How many revolutions will a pinion of 20 teeth make for 1 of a 
wheel with 125 ? 

125 -i- 20 = 6*25, or 6^ revolutions. 

Intermediate wheels, of whatever diameters, so as to connect 
communication at any required distance apart, cause no variation 
of velocity more than otherwise would result were the first and last 
in immediate contact. 

Rule. — To find tlie number of revolutions of the last, to 1 of the 
first, in a train of wheels and pinions. — Divide the product of all 
the teeth in the driving, by the product of all the teeth in the 
driven, and the quotient equal the ratio of velocity required. 

Required the ratio of velocity of the last, to 1 of the first, in the 
following train of wheels and pinions ; \\z., pinions driving, — the 
first of which contains 10 teeth, the second 15, and third 18 ; — 
wheels driven, — first 15 teeth, second 25, and third 32. 

1 r y Qg y qo = *225 of a revolution the wheel will make to 1 
of the pinion. 

A wheel of 42 teeth giving motion to one of 12, on which shaft 
is a pulley of 21 inches diameter, driving one of 6 ; required the 
number of revolutions of the last pulley to 1 of the first wheel. 

42 X 21 
-ioy^a = 12-25, or 12 J revolutions. 

Where increase or decrease of velocity is required to be commu- 
nicated by wheel-work, it has been demonstrated that the number 
of teeth on each pinion should not be less than 1 to 6 of its wheel, 
unless there be some other important reason for a higher ratio. 

Rule. — When time must he regarded. — Multiply the diameter,' 
or number of teeth in the driver, by its velocity in any given time, 
and divide the product by the required velocity of the driven ; the 
quotient equal the number of teeth, or diameter of the driven, to 
produce the velocity required. 

If a wheel containing 84 teeth makes 20 revolutions per minute, 
how many must another contain to work in contact, and make 60 
revolutions in the same time ? 



C0XTIXU0U3 CIRCULAR MOTION. 433 

84 X 20 ^^ 

— g^— = 28 teeth. 

From a shaft making 45 revolutions per minute, and with a pinion 

9 inches diameter at the pitch line, I wish to transmit motion at 15 

revolutions per minute ; what at the pitch line must be the diameter 

of the wheel ? 

45 X 9 ^, . , 
— Tc — = 27 inches. 

Required the diameter of a pulley to make 16 revolutions in the 
same time as one of 24 inches making 36. 

24 X 36 

— H-ft — — 54 inches. 

Rule. — The distance between the centres and velocities of two 
wheels being given, to find their proper diameters. — Divide the 
greatest velocity by the least ; the quotient is the ratio of diameter 
the wheels must bear to each other. Hence, divide the distance 
between the centres by the ratio plus 1 ; the quotient equal the 
radius of the smaller wheel ; and subtract the radius thus obtained 
from the distance between the centres ; the remainder equal the 
radius of the other. 

The distance of two shafts from centre to centre is 50 inches, 
and the velocity of the one 25 revolutions per minute, the other is 
to make 80 in the same time ; the proper diameters of the wheels 
at the pitch lines are required. 

50 

80 -r- 25 = 3*2, ratio of velocity, and o.^^ , -. = 11*9, the ra- 
dius of the smaller wheel ; then 50 — 11-9 = 38-1, radius of larger ; 
their diameters are 11*9 X 2 = 23-8, and 38*1 X 2 = 76-2 inches. 

To obtain or diminish an accumulated velocity by means of 
wheels and pinions, or wheels, pinions, and pulleys, it is necessary 
that a proportional ratio of velocity should exist, and which is 
simply thus attained : — Multiply the given and required velocities 
together, and the square root of the product is the mean or propor- 
tionate velocity. 

Let the given velocity of a wheel containing 54 teeth equal 16 
revolutions per minute, and the given diameter of an intermediate 
pulley equal 25 inches, to obtain a velocity of 81 revolutions in a 
machine ; required the number of teeth in the intermediate wheel, 
and diameter of the last pulley. 

\/81 X 16 == 36 mean velocity. 

54 X 16 ^, , 25 X 36 ^^ 

■ — og — == 24 teeth, and — g^ — — H'l inches, diameter of 

pulley. 

To determine the proportion of wheels for screw cutting by a 
lathe. — In a lathe properly adapted, screws to any degree of pitch, 
or number of threads in a given length, may be cut by means of a 

28 



434 THE PRACTICAL MODEL CALCULATOR. 

leading screw of any given pitch, accompanied with change wheels 
and pinions ; course pitches being effected generally by means of 
one wheel and one pinion with a carrier^ or intermediate tvheel, 
which cause no variation or change of motion to take place : hence 
the following 

Rule. — Divide the number of threads in a given length of the 
screw which is to be cut, by the number of threads in the same 
length of the leading screw attached to the lathe ; and the quotient 
is the ratio that the wheel on the end of the screw must bear to 
that on the end of the lathe spindle. 

Let it be required to cut a screw with 5 threads in an inch, the 
leading screw being of J inch pitch, or containing 2 threads in an 
inch ; what must be the ratio of wheels applied ? 

5 -T- 2 = 2*5, the ratio they must bear to each other. 

Then suppose a pinion of 40 teeth be fixed upon for the spindle, — 

40 X 2*5 = 100 teeth for the wheel on the end of the screw. 

But screws of a greater degree of fineness than about 8 threads 
in an inch are more conveniently cut by an additional wheel and 
pinion, because of the proper degree of velocity being more effec- 
tively attained ; and these, on account of revolving upon a stud, 
are commonly designated the stud-wheels, or stud-ivheel Sindpimon; 
but the mode of calculation and ratio of screw are the same as in 
the preceding rule ; — hence, all that is further necessary is to fix 
upon any 3' wheels at pleasure, as those for the spindle and stud- 
wheels, — then multiply the number of teeth in the spindle-wheel 
by the ratio of the screw, and by the number of teeth in that wheel 
or pinion which is in contact with the wheel on the end of the screw ; 
divide the product by the stud- wheel in contact with the spindle- 
wheel, and the quotient is the number of teeth requked in the wheel 
on the end of the leading screw. 

Suppose a screw is required to be cut containing 25 threads in 
an inch, the leading screw as before having 2 threads in an inch, 
and that a wheel of 60 teeth is fixed upon for the end of the spin- 
dle, 20 for the pinion in contact with the screw-wheel, and 100 for 
that in contact with the wheel on the end of the spindle ; — required 
the number of teeth in the wheel for the end of the leading 
screw. 

25 -^ 2 = 12-5, and J^~ = ^^^ *®^^^- 

Or, suppose the spindle and screw-wheels to be those fixed upon, 
also any one of the stud- wheels, to find the number of teeth in the 
other. 

60 X 12-5 ^^ ^ 60 X 12-5 x 20 ' ^^ 

150 X 100 = -^ *''*^' ""' 150 = 1^^ *^®*^- 



CONTINUOUS CIRCULAR MOTION. 



435 



Table of Change Wheels for Screw Cutting, the leading screw 
heing of J inch pitch, or containing two threads in an inch. 



2 . 

= 1 


Number of 
teeth ia 


o 

DO 




dumber of teeth 


la 


.5 

1 

5i 




tfumber of teeth 


in 


s> 


i 


<o 


ll 


■si 

11 


1 


i 


f? 


ll 


i 


o n 


1. 


bo . 


■si 


•3 

i 


.e-= 


gg 


1 


= i 


1 


•=l 


•si 


to . 


So 




c~ 


o o 






fl ^ 










a " 


'*'» 




52 


'ii 


.a _ 


iM 


25 


.25 


ll 


£•3 


j1 


Is 


.2 5 


'is 




i3 '^ 


^^ 


1- 


^^ 


P^ 


JL 


^ ^ 




r- 


1'^ 


•1'^ 


^^ 


1 


80 


40 


81 


40 


55 


20 


60 


19 


50 


95 


20 


100 


H 


80 


50 


^ 


90 


85 


20 


90 


19^ 


80 


120 


20 


130 


ij 


80 


60 


8| 


60 


70 


20 


75 


20 


60 


100 


20 


120 


If 


80 


70 


9i 


90 


90 


20 


95 


201 


40 


90 


20 


90 


2 


80 


90 


9| 


40 


60 


20 


65 


21 


80 


120 


20 


140 


2J 


80 


90 


10 


60 


75 


20 


80 


22 


60 


no 


20 


120 


2* 


80 


100 


m 


50 


70 


20 


75 


22^ 


80 


120 


20 


150 


2| 


80 


110 


11 


60 


65 


20 


120 


22| 


80 


130 


20 


140 


3 


80 


120 


12 


90 


90 


20 


120 


23| 


40 


95 


20 


100 


3J 


80 


130 


12| 


60 


85 


20 


90 


24 


65 


120 


20 


130 


3^ 


80 


140 


13 


90 


90 


20 


130 


25 


60 


100 


20 


150 


3f 


80 


150 


13^ 


60 


90 


20 


90 


25J 


30 


85 


20 


90 


4 


40 


80 


m 


80 


100 


20 


110 


26 


70 


130 


20 


140 


4i- 


40 


85 


14 


90 


90 


20 


140 


27 


40 


90 


20 


120 


4J 


40 


90 


14i- 


60 


90 


20 


95 


27J 


40 


100 


20 


110 


4| 


40 


95 


15 


90 


90 


20 


150 


28 


75 


140 


20 


150 


5 


40 


100 


16 


60 


80 


20 


120 


28J 


30 


90 


20 


95 


5i 


40 


110 


161 


80 


100 


20 


130 


30 


70 


140 


20 


150 


6 


40 


120 


16i 


80 


110 


20 


120 


32 


30 


80 


20 


120 


^ 


40 


130 


17 


45 


85 


20 


90 


33 


40 


110 


20 


120 


7 


40 


140 


17^ 


80 


100 


20 


140 


! 34 


30 


85 


20 


120 


7^ 


40 


150 


18 


40 


60 


20 


120 


I 35 


60 


140 


20 


150 


8 


30 


120 


18f 


80 


100 


20 


150 


' 36 


30 


90 


20 


120 



Table hg which to determine the Number of Teeth, or Pitch of 
Small Wheels. 



Diametral 


Circular 


Diametral 


Circular 


pitch. 


pitch. 


pitch. 


pitch. 


3 


1-047 


9 


•349 


4 


•785 


10 


•314 


5 


•628 


12 


•262 


6 


•524 


14 


•224 


7 


•449 


16 


•196 


8 


•393 


20 


•157 



Required the number of teeth that a wheel of 16 inches diameter 
will contain of a 10 pitch. 

16 X 10 = 160 teeth, and the circular pitch = '314 inch. 

What must be the diameter of a wheel for a 9 pitch of 126 teeth ? 



126 



= 14 inches diameter, circular pitch '349 inch. 



The pitch is reckoned on the diameter of the wheel instead of 
the circumference, and designated wheels of 8 pitch, 12 pitch, &c. 



436 



THE PRACTICAL MODEL CALCULATOR. 



Table of the Diameters of Wheels at their pitch circle, to contain 
a required number of teeth at a given pitch. 



I 


PITCH OF THE TEETH IN INCHES. 


1 in. 1 U li 1 If 1 li 11- 1 11 1* 1 2 in. 1 2J 1 2i 2i 2i | 3 in. 


a 1 


DIAMETER AT THE PITCH CIKCLE IN FEET AND INCHES. 


10 !o 3i0 3t0 4 1 


3 4i 


3 4§ 


5^ 5f iO 6 


64 


^ 6f 


7i 8 8f 


9f 


11 iO 3^0 4 


) 41 


3 5 


3 5f 


5i0 6i|0 6f 


7 


74 


7f|0 8|0 91 


lOf 


12 3^'0 4f 


3 4f 


3 51 


3 54 


6f|0 6110 74 


7f 


84 


8f 9f;0 lOf 


114 


13 


) 4i;0 41 


3 5i 


3 51 


6i 


6fi0 7f!0 7f 


8f| 


8f 


9f lOflo 114 


1 04 


14 


) 4iiO 5 


3 5f 


3 6i 


3 6J 


7f 


7f,0 84 


9 


94 


10 jO lU 1 Of! 


1 14 


15 


) 4^iO 51 


3 6 


3 6f 


7i 


7f 


84i0 9 


9f 


lOi 


lOfll 


1 U 


1 21 


16 


3 5^10 bi 


3 61 


3 7 


7f 


8f 


9 jO 9f 


OlOi 


lOf 


01141 Of 


1 2 


1 3f 


ir 


3 bM( 


) 6i 


3 61 


3 7i 


8i 


H 


9|:0 lOi 


OlOf 


114 


I Oil 141 


1 2f 


1 4i 


18 


[) 5|| 


) 6i 


3 7i 


3 8 


8f 


9f 


10 


3l0i 


0114 


1 Oi 


1 H 


1 2t 


1 31 


1 5i 


19 


[) 6 [ 


) 6^ 


7i 


3 81 


94 


9f 


OlOf 


111 


1 04 


1 Of 


i If 


1 34 


1 4f 


1 6i 


20 


61 


) 7* 


13 8 


[) 8f 


9f 


OlOf 


oiul 


1 


1 01 


1 14 


1 2f 


1 4 


1 54 


1 74 


21 


61 


) n 


81 


9i 


10 


Oil 


111 


1 04 


1 14 


1 21 


1 3 


1 41 


1 6f 


1 84 


22 


7 


3 7^ 


81 


9f 


OlOf 


0114 


1 Of 


1 14 


1 2 


1 2f 


1 31 


1 54 


1 n 


1 9 


23 


71 


} 8i 


94 


10 


11 


1 


1 Of 


1 11 


1 2f 


1 34 


1 44 


1 6| 


1 8 


] 10 


24 


7t 


9 8t 


9i 


OlOi 


Olli 


1 04 


1 14 


1 2f 


1 3f 


1 4i 


1 5i 


1 74 


1 9 


1 lOf 


25 


8 


9 


10 


Oil 


1 


1 1 


1 2 


1 2f 


1 3f 


1 4f 


1 6 


1 8 


1 9f 


lllf 


26 


8i 


9i 


OlOf 


om 


1 04 


1 14 


1 24 


1 34 


1 44 


1 54 


1 6f 


1 81 1 lOf 


2 Of 


27 


8f 


n 


lOS 


Ollf 


1 1 


1 2 


1 3 


1 44 


1 bl 


1 6i 


1 71 


1 94 


lllf 


2 11 


28 


9 


10 


Olli 


1 Oi 


1 u 


1 24 


1 3f 


1 4f 


1 bl 


1 6f 


1 8 


llOi 


2 04 


2 21 


29 


9i 


oiot 


llf 


1 01 


1 If 


1 3 


1 44 


1 5f 


1 64 


1 7f 


1 81 


1114 


2 If 


2 31 


30 


9i 


oios 


1 


1 14 


1 2f 


1 34 


1 44 


1 6 


1 74 


1 8i 


1 94 


2 


2 2i 


2 4f 


31 


9^ 


OlU 


1 Of 


1 If 


1 2f 


1 4 


1 5f 


1 64 


1 71 


1 9 


llOi 


2 Of 


2 34 


2 5f 


32 


om 


om 


1 Of 


1 2 


1 3f 


1 4f 


1 5f 


1 74 


1 8f 


1 9f 


111 


2 14 


2 4 


2 64 


33 


OlOi 


iH 


1 H 


1 2h 


1 31 


1 64 


1 64 


1 71 


1 9 


llOf 


lllf 


2 2i2 4^ 


2 74 


34 


om 


1 04 


1 If 


1 3 


1 4i 


1 5f 


1 7 


1 8f 


1 9f 


111 


2 Of 


2 3 12 5| 


2 84 


35 


om 


1 Of 


1 2 


1 31 


1 41 


1 64 


1 74 


1 9 


1104 


nil 


2 1 


2 3f!2 6f 


2 94 


36 


Olli 


1 1 


1 2f 


1 31 


1 5^ 


1 6f 


1 8 


1 94 


llOf 


2 Of 


2 2 


2 4fi2 74 


2 10f 


37 


111 


1 11 


1 21 


1 4i 


1 5f 


1 7i 


1 8f 


1 10 


1114 


2 1 


2 24 


2 542 8| 


2 llf 


38 


1 Oi 


1 If 


1 3i 


1 4f 


1 6i 


1 7f 


1 9i 


1 101 


2 Oi 


2 li 


2 3i 


2 6i 


2 91 


3 Oi 


39 


1 01 


1 2 


1 3^ 


1 5 


1 6f 


1 8i 


1 91 


nil 


2 Of 


2 2f 


2 4 


2 7 


2104 


3 U 


40 


1 01 


1 2f 


1 4 


1 5i 


1 71 


1 81 


llOf 


iiif 


2 14 


2 3 


2 4f 


2 7f 


2 101 


3 2i 


41 


1 1 


1 21 


1 4f 


1 6 


1 7f 


1 H 


llOf 


2 04 


2 24 


2 31 


2 5f 


2 8f 


2 llf 


3 34 


42 


1 If 


1 3 


1 41 


1 6f 


1 8 


1 91 


1114 


2 1 


2 21 


2 44 


2 6 


2 9t 


3 0| 


3 44 


43 


1 If 


1 3i 


1 54 


1 6^ 


1 8f 


1 lOi 


2 


2 If 


2 3f 


2 5 


2 65 


2 10ij? If 
2 11 3 24 


3 5 


44 


1 2 


1 3i 


1 5f 


1 n 


1 9 


llOi 


2 04 


2 2i 


2 4 


2 5S 


2 74 


3 6 


45 


1 21 


1 4i 


1 6 


1 71 


1 9411111 


2 1 


2 2f 


2 4f 


2 64 


2 H 


2II1I3 3| 


3 7 


46 


1 2f 


1 4i 


1 61 


1 84 


1 10 jl llf 


2 If 


2 34 


2 5i 


2 74 


2 9 


3 Of 13 4i 


3 7f 


47 


1 2^- 


1 4^ 


1 61 


1 8f 


1104 


2 Of 


2 If 


2 4 


2 6 


2 7f 


2 9f 


3 1413 54 


3 8f 


48 


1 3i 


1 5i 


1 74 


1 9 


1 11 


2 Of 


2 21 


2 4f 


2 64 


2 84 


2 10f 


3 2\ 


3 6 


3 9f 


49 


1 H 


1 5f 


1 7^ 


1 n 


1114 


2 If 


2 3f 


2 5i 


2 7i 


2 94 


2 11 


3 3 


3 6f 


3 lOf 


50 


1 3^ 


1 6 


1 8 


1 % 


2 


2 If 


2 3f 


2 5f 


2 7f 


2 91 


2 111 


3 31 


3 71 


3 111 


51 


1 4i 


1 6i 


1 81 


1101 


2 Of 


2 2t 


2 44 


2 64 


2 84 


2 104 


3 04 


3 44 


3 8f 


4 01 


52 


1 4i 


1 6f 


1 81 


llOi 


2 Of 


2 2f 


2 4f 


2 74 


2 94 


2 114 


3 U 


3 5f 


3 94 


4 If 


53 


1 4^ 


1 6i 


1 94 


llU 


2 11 


2 3§ 


2 5f 


2- 7f 


2 91 


2 llf 


3 2 


3 61 


3 104 


4 2f 


54 


1 5i 


1 7f 


1 n 


nil 


2 11 


2 3f 


2 6 


2 H 


2 101 


3 04 


3 2f 


3 7 


3 11 


4 34 


55 


1 b\ 


1 71 


1 H 


2 


2 2\ 


2 44 


2 6f 


2 8f 


2 11 


3 U 


3 34 


3 71 


4 04 


4 44 


56 


1 bi 


1 U 


1 lOi 


2 0^ 


2 21 


2 4f 


2 74 


2 9f 


2 111 


3 If 


3 44 


3 84 


4 ] 


4 54 


57 


1 6^ 


1 8^ 


1101 


2 0^ 


2 3^ 


2 5^ 


2 71 


2 10 


3 Oi 


3 24 


3 4i 


3 H 


4 If 


4 6f 


58 


1 6i 


1 81 


111 


2 11 


2 31 


\2 6 


2 8i 


2 101 


3 Of 


3 2,1 


3 5* 


3 104 


4 21 


4 7f 


59 


1 61 


1 n 


IIU 


2 1^ 


2 4i 


-2 6^ 


2 8f 


2 Hi 


3 14 


3 4 


3 6i 


3 114 


4 3t 


4 8f 


60 


1 7^ 


1 9t 


Ull^ 


2 21 


2 4t 


12 7 


2 9f 


2 111 


3 21 


3 41 


3 7 


3 HI 


4 44 


4 n 


61 


I 7^ 


1 9^ 


f2 0^ 


2 21 


2 5^ 


■2 74 


2 10 


3 01 


3 2f 


3 b\ 


3 71 


4 04 


4 5f 


4 m 


62 


1 71 


110^ 


^2 01 


2 3^ 


2 5t 


r2 8 


2104 


3 1 


3 34 


3 6 


3 84 


4 If 


4 6i 


4 1U 


63 


1 8 


llOJ 


^2 1 


2 3i 


-2 6 


2 81 


2 11 


3 11 


3 4^ 


3 61 


3 94 


4 24 


4 74 


6 04 


64 


1 8t 


IllOj 


^2 1^ 


2 4 


2 6^ 


^2 9^ 


2 llf 


3 21 


3 4| 


3 n 


3 9g 


4 3 


4 8 


5 14 


65 


1 81 


^1 Jl^ 


\2 n 


t2 4J 


t2 7 


2 91 


3 Oi 


3 2^ 


3 51 


3 8 


3 104 


4 31 


4 8f 


5 2 


66 


1 9 


nil 


\2 2\ 


t2 4^ 


^2 7J 


^2 10^ 


r3 0^ 


3 31 


3 6 


3 81 


3 1U 


4 44 


4 91 


5 3 


67 


1 9| 


12 


2 2t 


^2 5J 


^2 8 


2 101 


h3 If 


3 4 


3 6^ 


3 9i 


4 


4 5f 


4 10^ 


5 4 


68 


1 9> 


^2 Oj 


12 3 


2 5^ 


^2 8 


2 111 


3 1^ 


3 4t;3 7i 


3 10 


4 01 


4 64 


4114 


5 5 


69 


1 9| 


\2 Oi 


^2 3J 


12 6^ 


^2 85 


^2 111 


3 23 


3 5413 7f 


3 101 


4 14 


4 7 


5 0§ 


5 6 


70 


110il2 1 


2 3i 


12 6t 


f2 9j 


13 0^ 


3 3 


|3 6||3 84 


3 111 


4 24 


4 71 


5 U 


5 6f 



CONTINUOUS CIRCULAR MOTION. 



437 



'g 


PITCH OF THE TEETH IN INCHES 


1 


ll l'°-l 1* 1 


Li 11 U 1 H i 14 1 U 2in.| 21 2i 1 2i 1 2i | 3 in. 


I'i 


DIAMETER AT THE PITCH CIRCLE IN FEET AND INCHES. 


71110f2 li 2 


H 


2 7 I2 9l!3 01 


3 


H 


3 61 


3 9i 


4 





4 21 


4 8i 


5 21 


5 711 


72 110^2 im 


41 


2 7i2 10l|3 li 


3 


41 


3 61 


3 91 


4 


Of 


4 3i 


4 9i 


5 3 


5 81 


73 1 lU 


2 2i|2 


5 


2 8 2 10113 11 


3 


41 


3 7.V 


3 lOi 


4 


If 


4 4i 


4 10 


5 31 


5 n 


74 1 lU 


2 2.it2 


5i 


2 81 2 111 3 2i 


3 


5i 


3 71 


3 111 


4 


2 


4 5 


4 101 


5 41 


5 101 


75 1 111 


2 212 


51 


2 8II2 Ill's 2it 


3 


51 


3 8J 


3 111 


4 


2| 


4 51- 


4 111 


5 51 


5 111 


76 2 Oi 


2 3i 


2 


6i 


2 9i3 0i|3 3i 


3 


61 


3 91 


4 01 


4 


3-^. 


4 6i 


5 Oi 


5 6i 


6 Oi 


77 2 Oi 


2 3i 


2 


61 


2 9|j3 0|;3 31 


3 


61i3 91 


4 1 


4 


4 


4 71 


5 li 


5 71 


6 li 


78 2 01 


2 31 


2 


7 


2 101I3 u\i 41 


3 


7i 


3 lOi 


4 114 


4| 


4 71 


5 2 


5 8i 


6 2i 


79 2 n 


2 44'2 


71 


2 10^13 1413 41 


3 


8 


3 111 


4 2i4 


^ 


4 8h 


5 21 


5 91 


6 3i 


80 2 H 


2 41 


2 


71 


2 11 J3 2i{3 51 


3 


8i 


3 11« 


4 3 


4 


61 


4 9i 


5 31 


5 10 


6 41 


812 11 


2 5 


2 


Si 


2 IH 3 21 3 51 


3 


91 


4 01 


4 H 


4 


61 


410 


5 4i 


5 101 


6 51 


82 2 2* 


2 6g 


2 


81 


2 1113 313 61 


3 


91 


4 01 


4 4i 


4 


n 


4 10i 


5 5i 


5 HI 


6 61 


83 2 2i 


2 51 


2 


9 


3 013 3113 61 


3 


lOi 


4 H 


4 41 


4 


81 


4 Hi 


5 6 


6 01 


6 7i 


84,2 21 


2 6 


2 


91 


3 0|j3 4 J3 7i 


3 


101 


4 21 


4 5i4 


81 


5 01 


5 61 


6 li 


6 81 


852 3 


2 61 


2 


9i 


3 li 


3 4^;3 71 


3 Hi 


4 2J 


4 614 


n 


5 01 


5 71 


6 21 


6 91 


862 3f 


2 61 


2 


lOi 


3.11 


3 5il3 8i 


3 111 


4 3i 


4 61 


4 


101 


5 U 


5 8i 


6 3i 


6 10i 


87!2 3§ 


2 71 


2 101 


3 2 


3 5i3 9 




Oi 


4 31 


4 71 


4 101 


5 2i 


5 9i 


6 41 


6n 


88'2 4 


2 7i 


2 11 


3 2i 


3 6 3 9i 




1 


4 4i 


4 8 


4 m 


5 3 


5 10 


6 5 


7 


89;2 4f 


2 71 


2 111 


3 21 


3 6i 3 10 




u 


4 51 


4 81 


5 


Oi 


5 3i 


5 101 


6 51 


7 1 


90:2 4t 


2 8i 


2 


m 


3 3i 


3 7 3 10 i 




21 


4 51 


4 9i 


5 


01 


5 4i 


5 111 


6 6i 


7 2 


912 41 


2 8i 


3 


Oi 


3 31 


3 7i!3 11 




21 


4 6i 


4 91 


5 


li 


5 51 


6 01 


6 71 


7 21 


92^2 5i 


2 81 


3 


01 


3 4i 


3 71I3 111 




3i 


4 7 


4 10i 


5 


21 


5 51 


6 1 


6 Si 


7 31 


93 2 5f 


2 9i 


3 


1 


3 41 


3 81i4 01 




31 


4 7J. 


4 Hi 


5 


21 


5 61 


6 2 


6 91 


7 41 


94 2 51 


2 91 


3 


If 


3 51 


3 8114 01 




41 


4 81 


41115 


3i 


5 71 


6 21 


6 lOi 


7 51 


95 2 6i 


2 10 


3 


11 


3 51!3 9l!4 U 




41 


4 8i 


5 0ii5 


4i 


5 8 


6 3i 


6 111 


7 61 


96 2 6i 


2 10113 


21 


3 6 


3 9i|4 11 




oi 


4 91 


5 11|5 


5 


5 8S 


6 41 


7 


7 71 


97 2 61 


2 101 


3 


21 


3 6i 


3 10i:4 21 




6 


4 10 


5 115 


51 


5 9i 


6 5i 


7 01 


7 81 


98 2 7i 


2 11 


3 


3 


3 61 


3 104|4 21 




6i 


4 10i 


5 215 


6i 


5 101 


6 6 


7 If 


7 9i 


99 2 7i 


2 lit 


3 


31 


3 71 


3 Hi 


4 3i 




71 


4 11 


5 3 5 


7 


5 11 


6 61 


7 21 


.7 10i 


100 2 71 


2 111 


3 


31 


3 71 


3 111 


4 3i 




71 


4 111 


5 315 


n 


5 111 


6 7i 


7 31 


7 Hi 


1012 8i 


3 01 


3 


41 


3 8i 


4 Oi 


4 4i 




8i 


5 Oi 


5 4i5 


8i 


6 Oi 


6 81 


7 41 


8 Oi 


102 2 8i|3 Oi 


3 


4i 


3 81 


4 01 


4 41 




81 


5 1 


5 5 5 


9 


6 1 


6 91 


7 5i 


8 U 



Table of the Strength of the Teeth of Cast Iro7i Wheels at a 
given velocity. 



Pitch 
of teeth 


Thickness 
of teeth 


Breadth 
of teeth 


strength of teeth 


u horse power, at j 


3 feet per 


4 feet per 


6 feet per 


8 feet per i 


in inches. 


in inches. 


in inches. 


second. 


second. 


second. 


second. 


3-99 


1-9 


7-6 


20-57 


27-43 


41-14 


54-85 


3-78 


1-8 


7-2 


17-49 


23-32 


34-98 


46-64 


3-57 


1-7 


6-8 


14-73 


19-65 


29-46 


39-28 


3-36 


1-6 


6-4 


12-28 


16-88 


24-56 


32-74 


3-15 


1-5 


6 


10-12 


18-50 


20-24 


26-98 


2-94 


1-4 


5-6 


8-22 


10-97 


16-44 


21-92 


2-78 


1-3 


5-2 


6-58 


8-78 


13-16 


17-54 


2-52 


1-2 


4-8 


5-18 


6-91 


10-36 


18-81 


2-31 


11 


4-4 


3-99 


5-82 


7-98 


10-64 


2-1 


10 


4 


3-00 


4-00 


6-00 


8-00 


1-89 


•9 


3-6 


2-18 


2-91 


4-36 


5-81 


1-68 


•8 


3-2 


1-53 


2-04 


3-06 


3-08 


1-47 


•7 


2-8 


1-027 


1-87 


2-04 


2-72 


1-26 


•6 


2-4 


-64 


-86 


1-38 


1-84 


105 


•6 


2 


•375 


•50 


•75 


1-00 



438 THE PRACTICAL MODEL CALCULATOR. 



ADDITIONAL EXAMPLES ON THE VELOCITY OF WHEELS, 
DEUMS, PULLEYS, ETC. 

If a wheel that contains 75 teeth makes 16 revolutions per 
minute, required the number of teeth in another to "work in it, and 
make 24 revolutions in the same time. 

75 X 16 ^^ ' 
— 24 — = 50 teeth. 

A wheel, 64 inches diameter, and making 42 revolutions per 

minute, is to give motion to a shaft at the rate of 77 revolutions 

in the same time : required the diameter of a wheel suitable for 

that purpose. 

64 X 42 „, ^ . , 
— j— — = 34-9 inches. 

Required the number of revolutions per minute made by a wheel 
or pulley 20 inches 'diameter, when driven by another of 4 feet di- 
ameter, and making 46 revolutions per minute. 

48 X 46 

— 20 — ~ 110-4 revolutions. 

A shaft, at the rate of 22 revolutions per minute, is to give mo- 
tion, by a pair of wheels, to another shaft at the rate of 15J ; the 
distance of the shafts from centre to centre is 45J inches ; the di- 
ameters of the wheels at the pitch lines are required. 

45-5 X 15-5 . , 

22 4- 1^-^ ~ 18-81 radius of the driving wheel. 

45*5 X 22 
And Qo r 15.5 = 26*69 radius of the driven wheel. 

Suppose a drum to make 20 revolutions per minute, required the 
diameter of another to make 58 revolutions in the same time. 

58 -r- 20 = 2-9, that is, their diameters must be as 2*9 to 1 ; 
thus, if the one making 20 revolutions be called 30 inches, the 
other will be 30 — 2*9 = 10*345 inches diameter. 

Required the diameter of a pulley, to make 12J revolutions in 
the same time as one of 32 inches making 26. 

32 X 26 

— -.Q.g = 66-^6 inches diameter. 

A shaft, at the rate of 16 revolutions per minute, is to give mo- 
tion to a piece of machinery at the rate of 81 revolutions in the 
same time ; the motion is to be communicated by means of two 
wheels and two pulleys with an intermediate shaft; the driving 
wheel contains 54 feet, and the driving pulley is 25 inches diameter ; 
required the number of teeth in the other wheel, and the diameter 
of the other pulley. 



VELOCITY OF WHEELS, ETC. 



439 



36, the mean velocity between 16 and 81 



then, 
36 X 25 ' _ _ . , 
— ^Ri ~ ll'll inches, diameter oi 



v/81 X 16 

16x54 ^, 

o^ = 24 teeth ; and 

pulley. 

Suppose in the last example the revolutions of one of the wheels 
to be given, the number of teeth in both, and likewise the diameter 
of each pulley, to find the revolutions of the last pulley. 

= 36, velocity of the intermediate shaft ; 



and 



24 
36 X 25 



11-11 



81, the velocity of the machine. 



Tabl'S for finding the radius of a wheel when the pitch is giveji, or 
the pitch of a wheel when the radius is given^ that shall contain 
from 10 to 150 teeth^ and any pitch required. 



Number 
of Teeth. 


Eadius. 


Number 
of Teeth. 


Eadius. 


Number 
of Teeth. 


Radius. 


Number 
of Teeth. 


Kadius. i 

18-464 1 


10 


1-618 


46 


7-327 


81 


12-895 


116 


11 


1-774 


47 


7-486 


82 


13-054 


117 


18-623 1 


12 


1-932 


48 


7-645 


83 


13-213 


118 


18-782 


13 


2-089 


49 


7-804 


84 


13-370 


119 


18-941 


14 


2-247 ! 


50 


7-963 


85 


13-531 


120 


19-101 


15 


2-405 


51 


8-122 


86 


13-690 


121 


19-260 


16 


2-563 


52 


8-281 


87 


13-849 


122 


19-419 


17 


2-721 


53 


8-440 


88 


14-008 


123 


19-578 


18 


2-879 


54 


8-599 


89 


14-168 


124 


19-737 


19 


3-038 


55 


8-758 


90 


14-327 


125 


19-896 


20 


3-196 


56 


8-917 


91 


14-486 


126 


20-055 


21 


3-355 


57 


9-076 


92 


14-645 


127 


20-214 


22 


3-513 


58 


9-235 


93 


14-804 


128 


20-374 


23 


3-672 


59 


9-394 


94 


14-963 


129 


20-533 


24 


3-830 


60 


9-553 


95 


15-122 


130 


20-692 


25 


3-989 


61 


9-712 


96 


15-281 


131 


20-851 


26 


4-148 


62 


9-872 


97 


15-440 


132 


21-010 


27 


4-307 


63 


10-031 


98 


15-600 


133 


21-169 


28 


4-465 


64 


10-190 


'99 


15-759 


134 


21-328 


29 


4-624 


65 


10-349 


100 


15-918 


135 


21-488 


30 


4-788 


66 


10-508 


101 


16-077 


136 


21-647 


31 


4-942 


67 


10-667 


102 


16-236 


137 


21-806 


32 


5-101 


68 


10-826 


103 


16-395 


138 


21-9G5 


33 


5-260 


69 


10-985 


-104 


16-554 


139 


22-124 ; 


34 


5-419 


1 70 


11-144 


105 


16-713 


140 


22-283 1 


35 


5-578 


71 


11-303 


106 


16-873 


141 


22-442 


36 


5-737 


72 


11-463 


107 


17-032 


142 


22-602 


37 


5-896 


73 


11-62^ 


108 


17-191 


143 


22-761 


38 


6-055 


74 


11-781 


109 


17-350 


144 


22-920 


39 


6-214 


76 


11-940 


110 


17-509 


145 


23-079 


40 


6-373 


76 


12-099 


111 


17-668 


146 


23-238 


41 


6-532 


77 


12-258 


112 


17-827 


147 


23-397 


42 


6-691 


78 


12-417 


113 


17-987 


148 


23-556 


43 


6-850 


79 


12-576 


114 


18-146 


149 


23-716 


44 


7-009 


80 


12-735 


115 


18-305 


150 


23-875 


45 


7-168 


1 













Rule. — Multiply the radius in the table by the pitch given, and 
the product will be the radius of the wheel required. 



440 



THE PRACTICAL MODEL CALCULATOR. 



Or, divide the radius of the wheel by the radius in the table, 
and the quotient will be the pitch of the w^heel required. 

Required the radius of a wheel to contain 64 teeth, of 3 inch 
pitch. 

10-19 X 3 = 30-57 inches. 

What is the pitch of a wheel to contain 80 teeth, when the radius 
is 25-47 inches ? 

25-47 -^ 12-735 = 2 inch pitch. 

Or. set off upon a straight line AB seven times the pitch AC 
given ; divide that, or another exactly the same length, into eleven 
equal parts ; call each of those divisions four, or each of those di- 
visions will be equal to four teeth upon the radius. If a circle be 
made with any number (20) of these equal parts as radius, AC the 
pitch will go that number (20) of times round the circle. 




Were it required to find the diameter of a wheel to contain 17 
teeth, the construction would be as follows : — 



A 


1 C 


2 


3 






4 


5 






6 




7 


B 




1 


2 


3 


4 


5 


6 • 


7 


8 


9 


,'', 


n 1 


a 


1 i 1 


1 1 1 


I 1 1 


1 1 1 


1 1 
b 


1 


1 1 1 


1 I 1 


1 1 




1 i 1 


I 1 1 


1 1 1 


1 



24 --^8 



Thus, 4 divisions and J of another equal the radius of the wheel, 
that is a^ h^ = ab, and A^ C^ = AC. 



1 



VELOCITY OF WHEELS, ETC. 



441 



Regular approved proportions for wheels with flat arms in the 
middle of the ring, and ribs or feathers on each side. — The length 
of the teeth = f the pitch, besides clearance, or f the pitch, clear- 
ance included. 

Thickness of the teeth | the pitch. 

Breadth on the face.... 2| — 

Edge of the rim | — 

Rib projecting inside the rim | — 

Thickness of the flat arms | — • 

Breadth of the arms at the points = 2 teeth and J the pitch, 
getting broader towards the centre of the wheel in the proportion 
of J inch to every foot in length. 

Thickness of the ribs, or feathers, \ the pitch. 

Thickness of metal round the eye, or centre, | the pitch. 

Wheels made with plain arms, the teeth are in the same propor- 
tion as above ; the ring and the arms are each equal to one cog or 
tooth in thickness, and the metal round the eye same as above, in 
feathered wheels. 

These proportions differ, though slightly, in different works and 
in different localities ; but they are the most commonly employed, 
and are besides the most consistent with good and accurate work- 
manship. For the 
sake of more easy 
reference, we col- 
lect them into a I* 
table, which the 
annexed diagram 
will serve fully to 
explain. They 

stand thus : — ^_ 

= 1 pitch. 




ah = Pitch of teeth 
mn — Depth to pitch line, PP, 
ns ■]- nm = Working depth of tooth, 
Qh — ns = Bottom clearance, 

fh = Whole depth to root, 
pq = Thickness of tooth, 
rp = Width of space. 

The use of the following table is very evident, and the manner 
of applying it may be rendered still more obvious by the following 
examples : — 

rt = 3-1416. 

1. Given a wheel of 88 teeth, 2|- inch pitch, to find the diameter 
of the pitch circle. Here the tabular number in the second column 
answering to the given pitch is -7958, which multiplied by 88 gives 
70*03 for the diameter required. 

2. Given a wheel of 5 feet (60 inches) diameter, 2f inch pitch, 
to find the number of teeth. Here the factor in the third column 



442 



THE PRACTICAL MODEL CALCULATOR. 



pitch is 
60 gives 



corresponding to the given 
1-1333, which multiplied by 
68 for the number of teeth. 

It may, however, so happen that the 
answer found in this manner contains a 
fraction — which being inadmissible by 
the nature of the question, it becomes 
necessary to alter slightly the diameter 
of the pitch circle. This is readily ac- 
complished by taking the nearest whole 
number to the answer found, and find- 
ing the modified diameter by means of 
the second column. The following case 
will fully explain what is meant : 

3. Given a wheel 33 inches diameter, 
If inch pitch, to find the number of 
teeth. The corresponding factor is 
1-7952, which multiplied by 33 gives 
59*242 for the number of teeth, that is, 
59^ teeth nearly. Now, 59 would here 
be the nearest whole number ; but as a 
wheel of 60 teeth may be preferred for 
convenience of calculation of speeds, we 
may adopt that number and find the di- 
ameter corresponding. The factor in 
the second column answering to If pitch 
is -557, and this multiplied by 60 gives 
33*4 inches as the diameter which the 
wheel ought to have. 

EuLE. — To find the power that a cast iron wheel is capable of 
transmitting at any given velocity. — Multiply the breadth of the 
teeth, or face of the wheel, in inches, by the square of the thick- 
ness of one tooth, and divide the product by the length of the teeth, 
the quotient is the strength in horse power at a velocity of 136 
feet per minute. 

Required the power that a wheel of the following dimensions 

ought to transmit with safety, namely, 

Breadth of teeth 7 J inches, 

Thickness c 1*4 

And length 2 

1-42 = 1-96, and 7*5 x 1-96 ^ ^, , 

^ =7-35 horse power. 

The strength at any other velocity is found by multiplying' the 
power so obtained by any other required velocity, and by -</D44, 
the quotient is the power at that velocity. 

Suppose the wheel as above, at a velocity of 320 feet per minute. 
7-35 X 320 X -0044 = 10-3488 horse power. 



Pitch in 

inches and 

parts of an 

inch. 


IT 


N = pXD 


Rule.— To find 
the diameter m 
inches, multi- 
ply the number 
of teeth bj the 

ber answering 
to the given 


Rdle—To find 
the number of 
teeth, mul tip! v| 
the given dia-l 
meter is inches 1 
by the tabular! 
number an-l 
svrering to the 
given pitch. 


Values of P 


Values of - 


Values Of J 


6 


1-9095 , 


•5236 


5 


1-5915 


•6283 


4J 


1-4270 


•6981 


4 


1-2732 


•7854 


3^ 


1-1141 


•8976 


3 


•9547 


1-0472 


23. 


-8754 


1-1333 


2A^ 


•7958 


1-2566 


91 


•7135 


1-3963 


2 


-6366 


1-5708 


1| 


-5937 


1-6755 




•5570 


1-7952 ! 


If 


•5141 


1-9264 ! 


\x 


•4774 


2-0944 i 


If 


•4377 


2-2848 1 


H 


•3979 


2-5132 1 


H 


•3568 


2-7926 




•3183 


3-1416 ! 


|. 


-2785 


3-5904 i 


|. 


•2387 


4-1888 j 


1 


•1989 


5-0266 




•1592 


6-2832 1 


3. 


•1194 


8-3776 ! 


1 


-0796 


12-5664 \ 



MAXIMUM VELOCITY AND POWER OF WATER WHEELS. 443 



ON THE MAXIMUM VELOCITY AND POWER OF WATER 
WHEELS. 

OF UNDERSHOT WHEELS. 

The term "undershot" is applied to a wheel when the water 
strikes at, or below, the centre ; and the greatest effect is produced 
when the periphery of the wheels moves with a velocity of '57 that 
of the water ; hence, to find the velocity of the water, multiply the 
square root or the perpendicular height of the fall in feet by 8, 
and the product is the velocity in feet per second. 

Required the maximum velocity of an undershot wheel, when 
propelled by a fall of water 6 feet in height. 

v/6 = 2-45 X 8 = 19-6 feet, velocity of water. 
And 19*6 X '57 = 11-17 feet per second for the wheel. 

OF BREAST AND OVERSHOT WHEELS. 

Wheels that have the water applied between the centre and the 
vertex are styled breast wheels, and overshot when the water is 
brought over the wheel and laid on the opposite side ; however, in 
either case the maximum velocity is f that of the water ; hence, 
to find the head of water proper for a wheel at any velocity, say : 

As the square of 16-083, or 258-67, is to 4, so is the square of 
the velocity of the wheel in feet per second to the head of water 
required. By head is understood the distance between the aper- 
ture of the sluice and where the water strikes upon the wheel. 

Required the head of water necessary for a wheel of 24 feet 
diameter, moving with a velocity of 5 feet per second. 

— 2~ = 7*5 feet, velocity of the water. 
And 258-67 ; 4 : : 7-5^ : -87 feet, head of water required. 

But one-tenth of a foot of head must be added for every foot 
of increase in the diameter of the wheel, from 15 to 20 feet, and 
•05 more for every foot of increase from 20 to 30 feet, commencing 
with five-tenths for a 15 feet wheel. 

This additional head is intended to compensate for the friction 
of water in the aperture of the sluice to keep the velocity as 3 to 2 
of the wheel ; thus, in place of -87 feet head for a 24 feet wheel, 
it will be -87 + 1-2 = 2-07 feet head of water. • 

If the water flow from under the sluice, multiply the square 
root of the depth in feet by 5-4, and by the area of the orifice also 
in feet, and the product is the quantity discharged in cubic feet 
per second. 

Again, if the water flow over the sluice, multiply the square 
root of the depth in feet by 5*4, and f of* the product multiplied 



444 THE PRACTICAL MODEL CALCULATOR. 

by the length and depth, also in feet, gives the number of cubic 
feet discharged per second nearly. 

Required the number of cubic feet per second that will issue 
from the orifice of a sluice 5 feet long, 9 inches wide, and 4 feet 
from the surface of the water. 

v/4 = 2 X 5-4 = 10-8 feet velocity. 
And 5 X -75 X 10-8 = 40-5 cubic feet per second. 

What quantity of water per second will be expended over a wear, 
dam, or sluice, whose length is 10 feet, and depth 6 inches ? 

1-20744 X 2 
x/-5 = -2236 X 5-4 = g = -80496 feet velocity. 

Then 10 x -5 = 5 feet, and -80496 x 5 = 4-0248 cubic feet 
per second nearly. 

In estimating the power of water wheels, half the head must be 
added to the whole fall, because 1 foot of fall is equal to 2 feet of 
head ; call this the effective perpendicular descent ; multiply the 
weight of the water per second by the effective perpendicular de- 
scent and by 60 ; divide the product by 33,000, and the quotient 
is the effect expressed in horse power. 

Given 16 cubic feet of water per second, to be applied to an under- 
shot wheel, the head being 12 feet ; required the power produced. 

6x16x62-5 x60 ^^^ , 
12 -i- 2 = 6 and ooaaa = 10*9 horse power nearly. 

Givem 1(5 cubic feet of water per second, to be applied to a high 
breast or an overshot wheel, with 2 feet head and 10 feet fall ; 
required the power. 

; 1 + 10 X 16 X 62-5 X 60 ^^ , 
2-7-2 = 1 and SSMO ~ horse power. 

Only about two-thirds of the above results can be taken as real 
communicative power to machinery. 

OF THE CIRCLE OP GYRATION IN WATER WHEELS. 

The centre or circle of gyration is that point in a revolving body 
into which, if the whole quantity of matter were collected, the 
same moving force would generate the same angular velocity, 
which renders it of the utmost importance in the erection of water 
wheels, and the motion ought always to be communicated from that 
point when it is possible. 

Rule. — To find the circle of gyration. — Add into one sum twice 
the weight of the shrouding, buckets, &c., multiplied by the square 
of the radius, f of the weight of the arms, multiplied by the square 
of the radius, and the weight of the water multiplied by the square 
of the radius also ; divide the sum by twice the weight of the 
shrouding, arms, &c., added to the weight of the water, and the 
square root of the quotient is the distance of the circle of gyra- 
tion from the centre of suspension nearly. 



MAXIMUM VELOCITY AND POWER OF WATER WHEELS. 445 

Required tlie distance of the centre of gyration from the centre 
of suspension in a water wheel 22 feet diameter, shrouding, buckets, 
&c. = 18 tons, arms = 12 tons, and water = 10 tons. 

22 ~ 2 = 11 and IP = 121 

Then, 18 X 2 = 36 X 121 = 4356 

f ofl2 = 8 X 121 = 968 

water = 10 x 121 = 1210 

6534 



And 18 + 12 X 2 = 60 + 10 = TO ; hence, 

\/~frr = 9*6 feet from the centre of suspension nearly. 



Table of Angles for Windmill Sails. 



Niimhftr. 


Angle with the Plane of Motion. 


1 


18° 


24° 


2 


19 


21 


3 


18 


18 


4 


16 


14 


5 


12^ 


9 


6 


7 


3 extremity. 



The radius is supposed to be divided into six equal parts, and J 
from the centre is called 1, the extremity being denoted by 6. 

The first column contains the angles according to an old custom ; 
but experience has taught us that the angles in the second column 
are preferable. 



THE VELOCITY OF THRESHING MACHINES, 
IRON, ETC. 



MILLSTONES, BORING 



The drum or beaters of a threshing machine ought to move with 
a velocity of about 3000 feet per minute ; hence, divide 11460 by 
the diameter of the drum in inches ; or 955 by the diameter of the 
drum in feet; and the quotient is the number of revolutions 
required per minute. And the feeding rollers must make half 
the revolutions of the drum, when their diameters are about 
3J inches. 

If the machine is driven by horses, their velocity ought to be 
from 2J to 3 times round a 24 feet ring per minute. 

Divide 500 by the diameter of a millstone, in feet, or 6000 by 
the diameter in inches, and the quotient is the number of revolu- 
tions required per minute. 

In boring cast iron the cutters ought to have a velocity of about 
108 inches per minute, or divide 36 by the diameter in inches, the 
quotient is the number of revolutions of the boring head per minute. 
And divide 100 by the diameter in inches, the quotient is the num- 
ber of revolutions per minute, for turning wrought iron in general, 
and about half that velocity for cast iron. 



446 THE PRACTICAL MODEL CALCULATOR. 



OF PTJMPS AND PUMPING ENGINES. 

Pumps are chiefly designated by the names of lifting and force 
pumps ; lifting pumps are applied to wells, &c., where the height 
of the bucket, from the surface of the water, must not exceed 33 feet ; 
this being nearly equal to the pressure of the atmosphere, or the 
height to which water would be forced up into a vacuum by the 
pressure of the atmosphere. Force pumps are applicable on all 
other occasions, as raising water to any required height, supplying 
boilers against the force of the steam, hydrostatic presses, &c. 

The power required to raise water to any height is as the weight 
and velocity of the water with an addition of about | of the whole 
power for friction ; hence the 

Rule. — Multiply the perpendicular height of the water, in feet, 
by the velocity, also in feet, and by the square of the pump's 
diameter in inches, and again by -341 ; (this being the weight of 
a column of water 1 inch diameter, and 12 inches high, in lbs. 
avoirdupois;) divide the product by 33,000, and I of the quotient 
added to the whole quotient will be the number of horse power 
required. 

Required the power necessary to overcome the resistance and 
friction of a column of water 4 inches diameter, 60 feet high, and 
flowing with a velocity of 130 feet per minute. 

60 xl30 x42x -341 1-3 ^^ o . ., 

ooAAQ = -r- = '26 + 1-3 = 1d6 horse power nearly. 

Hot liquor pumps, or pumps to be employed in raising any fluid 
where steam is generated, require to be placed in the fluid, or as 
low as the bottom of it, on account of the steam filling the pipes, 
and acting as a counterpoise to the atmosphere ; and the diameter of 
the pipes to and from a pump ought not to be less than f of the 
pump's diameter. 

Rule. — The diameter of a pump and velocity of the ivater given, 
to find the quantity discharged in gallons, or cubic feet, in any 
given time. — Multiply the velocity of the water, in feet per minute, 
by the square of the pump's diameter in inches, and by -041 for 
gallons, or -0005454 for cubic feet, and the product will be the 
number of gallons, or cubic feet, discharged in the given time 
nearly. 

What is the number of gallons of water discharged per hour by 
a pump 4 inches diameter, the water flowing at the rate of 130 
feet per minute ? 

130 X 60 = 7800 feet per hour. 
And, 7800 x 4^ x -041 = 5116-8 gallons. 

Rule 1. — The length of stroke and number of strokes given, to 
find the diameter of a pump, and number of horse power that will 
discharge a given quantity of water in a given time. — Multiply the 



OF PUMPS AND PUMPING ENGINES. 447 

number of cubic feet by 2201, and divide the product by the ve- 
locity of the water, in inches, and the square root of the quotient 
will be the pump's diameter, in inches. 

2. Multiply the number of cubic feet by 62*5, and by the per- 
pendicular height of the water in feet, divide the product by 33,000, 
then will | of the quotient, added to the whole quotient, be the 
number of horse power required. 

Required the diameter of a pump, and number of horse power, 
capable of filling a cistern 20 feet long, 12 feet wide, and 6J feet 
deep, in 45 minutes, whose perpendicular height is 53 feet ; the 
pump to have an effective stroke of 26 inches, and make 30 strokes 
per minute. 

20 X 12 X 6-5 = 1560 cubic feet, and 

1560 

-jr-- = 34-66 cubic feet per minute. 

Then, 34-66 x 2201 , ,, . , ,. 

/26 X SO ~ inches diameter of pump. 

And 34-66 x 62-5 x 53 3-48 ^^ , ^ .o . .n r. 

^^^^ = — = .69 + 3-48 = 4-17 horse 

power. 

Rule. — To find the time a cistern will take in filling, when a 
known quantity of water is going in, and a known portion of that 
water is going out, in a given time. — Divide the content of the cis- 
tern, in gallons, by the difference of the quantity going in, and 
the quantity going out, and the quotient is the time in hours and 
^arts that the cistern will take in filling. 

If 30 gallons per hour run in and 22J gallons per hour run out 
of a cistern capable of containing 200 gallons, in what time will 
the cistern be filled ? 

30 - 22-5 = 7-5, and 200 -f- 7-5 = 26'66Q, or 26 hours and 
40 minutes. 

To find the time a vessel will take in emptying itself of water. — 
Mr. O'Neill ascertained, from very accurate experiments, that a 
vessel, 3-166 feet long and 2*705 inches diameter, would empty it- 
self in 3 minutes and 16 seconds, through an orifice in the bottom, 
whose area is -0141 inches ; and another 6-458 feet long, the dia- 
meter and orifice, as before, would do the same in 4 minutes and 40 
seconds ; hence, from these experiments, a rule is obtained, namely, 

Multiply the square root of the depth in feet by the area of the 
falling surface in inches, divide the product by the area of the ori- 
fice, multiplied by 3-7, and the quotient is the time required in 
seconds, nearly. 

How long will it require to empty a vessel of water, 9 feet high, 
and 20 inches diameter, through a hole f inch in diameter ? 
\/9 = 3, the square root of the depth, 
314-16 inches, area of the falling surface, 
•4417 inches, area of the orifice ; 



448 THE PRACTICAL MODEL CALCULATOR. 

Then, 314-16 x 3 ^^^ ^ 

•4417 X S-7 ~ ^*^'" seconds, or 9 minutes and 36 seconds. 

On the pressure of fluids. — The side of any vessel containing a 
fluid sustains a pressure equal to the area of the side, multiplied by 
half the depth ; thus, 

Suppose each side of a vessel to be 12 feet long and 5 feet deep, 
when filled with water, what pressure is upon each side ? 
12 X 5 = 60 feet, the area of the side, 
2*5 feet = half the depth, and 
62-5 lbs. = the weight of a cubic foot of water. 
Then, 60 x 2-5 x 62-5 = 93T5 lbs. 
Rule. — To find the weight that a given power can raise hy a 
hydrostatic press. — Multiply the square of the diameter of the ram 
in inches by the power applied in lbs., and by the effective leverage 
of the pump-handle ; divide the product by the square of the pump's 
diameter, also in inches, and the quotient is the weight that the 
power is equal to. 

What weight will a power of 50 lbs. raise by means of a hydro- 
static press, whose ram is 7 inches diameter, pump J, and the ef- 
fective leverage of the pump-handle being as 6 to 1 ? 

72 X 50 X 6 

:gY^i = 19200 lbs., or 8 tons 11 cwt. 

In the following rules for pumping engines the boiler is supposed 
- to be loaded with about 2|^ lbs. per square inch, and the barometer 
attached to the condenser indicating 26 inches on an average, or 
13 lbs., = 15^ lbs., from which deduct ^ for friction, leaves a pres- 
sure of 10 lbs. nearly upon each square inch of the piston. 

EuLE. — To find the diameter of a cylinder to work a pump of a 
given diameter for a given depth. — Multiply the square of the 
pump's diameter in inches by ^ of the depth of the pit in fathoms, 
and the square root of the product will be the cylinder's diameter 
in inches. 

Required the diameter of a cylinder to work a pump 12 inches 
diameter and 27 fathoms deep. 

>/(122 X 9) = 36 inches diameter. 

Rule. — To find the diameter of apump^ that a cylinder of a given 
diameter can tuorlc at a given depth. — Divide three times the square 
of the cylinder's diameter in inches by the depth of the pit in fa- 
thoms, and the square root of the quotient will be the pump's di- 
ameter in inches. 

What diameter of a pump will a 36-inch cylinder be capable of 
working 27 fathoms deep ? 



i 



eyj = 12 inches diameter. 



Rule. — To find the depth from which a pump of a given diameter 
will work hy means of a cylinder of a given diameter. — Divide three 



OF PUMPS AND PUMPING ENGINES. 449 

times the square of the cylinder's diameter in inches by the square 
of the pump's diameter also in inches, and the quotient will be the 
depth of the pit in fathoms. 

Required the depth that a cylinder of 36 inches diameter will 
work a pump of 12 inches diameter. 



J 



36^ X 3 ^^ ^ , 
~TXT~ — 27 fathoms. 



An inelastic body of 30 lbs. weight, moves with a 3 feet velo 
city, and is struck by another inelastic body having a 7 feet velo- 
city, the two will then proceed, after the blow, with the velocity 

_ 50 X 7 + 30 X 3 _ 350 + 90 _ 44 _ 11 _ 

^ ~ 50 + 30 ~ 80 - 8 ~ 2 ~ ^2 feet. 

To cause a body of 120 lbs. weight to pass from a velocity c^ = 
1 J feet into a 2 feet velocity v, it is struck by a heavy body of 50 
lbs., what velocity will the body acquire ? Here 

(v - eJ M, ^ (2 - 1-5) X 120 ^ 6 „ ^ 
e.-v-^ M; = 2 + ^ ^ = 2 + g = 3-2 

feet. 

Two perfectly elastic spheres, the one of 10 lbs. the other of 16 
lbs. weight, impinge with the velocities 12 and 6 feet against each 
other, what will be their velocities after impact ? Here M^ = 10 
and c^ = 12 feet, but M3 = 16 and c^ = — 6 feet, hence the loss 
of velocity of the first body will be 

2x16(12+6) 2 X 16 X 18 ^^_, , 
c^ — v^ = -|^Q I -^Q = 26 ^ 22-154 feet ; and 

" 2 X 10 X 18 
the gain in velocity of the other, % — C3 = ^ • = 13*846 

feet. From this the first body after impact will recoil with the ve- 
locity Vj = 12 — 22-154 = — 10-154 feet; and the other with 
that of — 6 + 13-846 = 7,846 feet. Moreover, the measure of 
vis viva of the two bodies after impact = M^v^^ + Mgv/ = 10 X 
10-1542 4. 16 X 7-846^ = 1031 + 985 = 2016, as likewise of that 
before impact, namely : M^c^^ + M^^/ = 10 x 12^ + 16 X 6^ = 
1440 + 576 = 2016. Were these bodies inelastic, the first would 

c — V 
only lose in velocity -^— « — * ~ 11-077 feet, and the other gain 

V "^ € 

-^— o — - = 6*923 feet ; the first would still retain, after impact, the 

velocity 12 — 11-077 = 0-923 feet, and the second take up the 
velocitv — 6 + 6-923 = 0-923, and the loss of mechanical effect 
would 'be (2016 - (10 + 16) 0-9232) h- 2^ = (2016 - 2-22) x 
0-0155 = 29-35 ft. lbs. 



450 



CENTRIPETAL AND CENTRIFUGAL FORCE. 

1. What is the centrifugal force of a body weighing 20 lbs. 
that describes a circle of 10 feet radius 200 times in a minute ? 

•000331 X 200^ X 20 X 10 = 2648 lbs., the centrifugal force. 
•00331 is a constant number. 

It is a well established fact that the centrifugal force is to the 
weight of the body as double the height due to the velocity is to 
the radius of revolution. Hence, this question may be thus solved ; 

20 X 3-1416 = 62-832, the circumference of the circle of 10 
feet radius. 

62-832 X 200 = 12566^4 feet, the space passed over by the 
weight in one minute. 

12566^4 

— ^ — = 209^44 feet, the space described in a second, which 

is called the velocity. 

(209^44)2 

— ^jTT — = 681 ^136 feet, the height due to the velocity. 

If F be the centrifugal force — 

F : 20 : : 1362^272 : 10. 

1362^272 X 20 
. •. F = ^ = 2724^544 lbs. The former rule gives 

2648 lbs. 

2. What is the centrifugal force at the equator on a body weigh- 
ing 300 lbs., supposing the radius of the earth = 21000000 feet, 
and the time of rotation = 8640(T'' = 24 hours ? 

21000000 X 300 ^ ^^_^ 
F = 1^224 X sQiOO^ ^ 1-03298 lbs., or one pound 

very nearly. 1^224 is a constant multiplier. 

3-1416 X 21000000 = 65973600 feet, } the circumference 

of the earth at the equator. 

2 X 65973600 ..^^.^ , , , . , , . , 
«7O7)0 ~ 1527^16 feet, the velocity of the weight. 

each second. 

(1527*16? 

g. . — = 36214-56, the height due to the velocity. 

F : 300 :: 72429*12 : 21000000. 

^ 72429-12 X 300 ^„ ^ , , , . 

F = 91QQQQQQ — ~ 1*034/ nearly, as by the former ap- 
proximate method. 

3. If a body weighing 100 lbs. describe a circle of 10 feet radius 
300 times a minute, what is the diameter of a cast iron cylindrical 



CENTRIPETAL AND CENTRIFUGAL FORCE. 



451 



rod, connecting the Body with the axis, that will safely support this 
weight ? The centrifugal force will be, 

•000331 X 3002 X 100 x 10 = 29790 lbs. 

From the strength of materials, page 281, we find that the ulti- 
mate cohesive strength for each circular inch of cross sectional area 
is 14652 lbs. ; but one-third of this weight, or 4884 lbs., can only 
be applied with safety. 

/29790 
AgQA ~ 2*46982 inches, the diameter of the cylindrical rod. 

4. The dimensions, the density, and strength of a millstone 
ABDE are given ; it is required to find the angular velocity v, in 
consequence of which rupture will take place on account of the 
centrifugal force. 



v^ 




If we put the radius of the millstone == r^ = 24 inches = 
CG; the radius = OK of its eye = r^ = 4 inches; the height 
PQ = GH = I = 12 inches ; the density = ^ = 2500 = specific 
gravity of the millstone ; and the modulus of strength = K = 
750 lbs. = the ultimate cohesive strength of each square inch of 
cross sectional area in the section PH, supposing the centrifugal 
forces — F and + F to cause the separation in this section. 
(^1 ~~ ^3) ^ — ^^®^ ^^ parallelogram GR. 
Hence, the force in lbs. required to cause rupture will be, 
2 (t-j — rj ? X K ; the weight of the stone G = « (r^^ — r/) ly, 
and the radius of gyration of each half of the stone, i. e. the distance 

4. f ^ ^_ f "J 
of its centre of gravity from the axis of rotation r = -q— X - ^ __ \ ' 

At the moment of rupture, the centrifugal force of half the stone 
is equivalent to the strength ; we hence obtain the equation of con- 



452 THE PRACTICAL MODEL CALCULATOR. 

p 7 

dition ^ X J — = 2 (r^ - r,) IK, i. e. "2 X f {r^' - r/) - = 
2 (t-j — rj ^K; or leaving out 2 1 on both sides, it follows that 
mg {r, ~ r,) K / 3^K ~~ 

If r, = 2 feet = 24 inches, r^ = 4 inches, K = 750 lbs., and 
the specific gravity of the millstone = 2*5 ; therefore the weight 

62-5 X 2-5 
of a cubic inch of its mass = — ^798 — ~ 0-0903 lbs. ; it follows 

that the angular velocity at the moment of rupture is, 

/ 3 X 12 X 32-2 X 750 /869400 

" = W 688 X 0-,9903 = ^6^1261 ^ ^^^'^ '''^^^^• 

If the number of rotations per minute = n, we have then w = 
2 rt 71 , . , 30 « , 30 X 112-1 
gQ ; hence, inversely, n =~~r~'> out here = — = 1070. 

The average number of rotations of such a millstone is only 120, 
therefore 9 times less. 

With what velocity must a body of 8 lbs. impinge against an- 
other at rest of 25 lbs., in order that the last may have a velocity 
of 2 feet ? Were the bodies inelastic, we should then have to put : 

^ "" M +M ' 1- ®- 2 = 8~+25' ^1 = -^ = ^i feet, the re- 

quired velocity ; but were they elastic, we should have v^ = j^ 'jV ; 

33 
hence, ^1 = "q~ = 4J feet. 

If in a machine, 16 blows per minute take place between two in- 
1000 1200 

elastic bodies M^ = lbs. and M^ = — — lbs., with the velo- 

cities c. = 5 feet, and c^ = 2 feet, then the loss in mechanical ef- 

16 (5 - 2f 1000-1200 
feet from these blows will be : L = ^q X — 2^ — x — 220O — ~ 

j^ X 9 X Q^ X -^ = 0-576 X -]jf = 20-94 units of work per 

second. 

If two trains upon a railroad of 120000 lbs. and 160000 lbs. 
weight, come into collision with the velocities c^ — 20, and c. = 
15 feet, there will ensue a loss of mechanical effect expended upon 
the destruction of the locomotives and carriages, which in the case 
of perfect inelasticity of the impinging parts, will amount to 

(2 H- 15y 120000 X 160000 _ J_ 1920000 _ 

= ^ 2^ ^ 280000 - ^5 X 64-4 ^ 28 

1344000 ft. lbs., or units of work. 



453 



SHIP-BUILDING AND NAVAL ARCHITECTirRE. 



Two rules, by which the principal calculations in the art of ship- 
building are made, may be employed to measure the area or super- 
ficial space enclosed by a curve, and a straight line taken as a base. 

Rule I. — If the area bounded by the curve line ABC and the 
straight line AC is required to be estimated, by the rule, the base 
AC is divided into an even number of equal parts, to give an odd 
number of points of division. 




8 9 10 11 12 13 14 15 16 17 18 19 20 210 



Where the base AC is divided into twenty equal parts, giving 
twenty-one points of division, and the lines I'l, 2*2, 3-3, &c., are 
drawn from these points at right angles or square to AC, to meet 
the curve ABC, these lines, 1-1, 2*2, 3*3, &c., are denominated or- 
dinates, and the linear measurement of them, on a scale of parts, 
is taken and used in the following general expression of the rule. 

Area = {A + 4P + 2Q} |. 

Where A = sum of the first and last ordinates, or 1*1 and 21*21. 

4 P = sum of the even ordinates multiplied by 4. 

Or, {2d 4- 4th -f 6th + 8th + 10th + 12th + 14th + 16th + 
18th -f 20th} X 4. 

2 Q = sum of the remaining ordinates ; or, 

{3d + 5th + 7th -f 9th -f 11th + 13th + 15th -f 17th -f 
19th} X 2. 

And r is equal to the linear measurement of the common inter- 
val between the ordinates, or one of the equal divisions of the base 
AC. This rule, for determining the area contained under the curve 
and the base, may be put under another form ; for as the 

T 

Area = {A + 4P + 2Q} x^; it may be transferred into 
A „ ^ 1 2r 



Area = J^+^^ + Q} 



The practical advantages to be derived from this modification of 
the general rule will appear when the methods of calculation are 
further developed. 



454 



THE PRACTICAL MODEL CALCULATOR. 




16 C 



Rule II. — If the base AC be so divided that the equal intervals 
are in number a multiple of the numeral 3, then the total number 
of the points of division, and consequently the ordinates to the 
curve, will be a multiple of the numeral 3 with one added, and the 
area under the curve ABC, and the base AC, can be determined 
by the following general expression : 

3r 

Area = {A + 2 P -f 3 Q} x -g. 

Where A = sum of the first and last ordinates, or 1 and 16. 

2 P = sum of the 4th, 7th, 10th, 13th, multiplied by 2, or ordi- 
nates bearing the distinction of being in position as multiples of 
the numeral 3, with one added. 

3 Q, the sum of the remaining ordinates, multiplied by 3, or of 
the 2d, 3d, 5th, 6th, 7th, 8th, 9th, 11th, 12th, 14th, and 15th, 
multiplied by 3. 

The number of equal divisions for this rule must be either 3, 6, 
9, 12, or 15, &c., being multiples of the numeral 3, whence the or- 
dinates will be in number under such divisions, multiples of the 
numeral 3, with one added. 

This rule admits also of a modification in form, to make it more 
convenient of application. 

3 
For area = {A + 2 P + 3 Q} x g r. 

As before advanced for the change adopted in the general ex- 
pression for the first rule, the utility of this modification of the 
second rule will be observable when the calculations on the im- 
mersed body are proceeded with. 

The rules are formed under the supposition that in the first rule 
the curve ABC, which passes through the extremities of the ordi- 
nates, is a portion of a common parabola, while in the second rule 
the curve is assumed to be a cubic parabola ; the results to be ob- 
tained from an indiscriminate use of either of these rules, differ 
from each other in so trifling a degree, (considered practically and 
not mathematically,) as not to sensibly affect the deductions derived 
by them. 

William O'Neill, or, as English writers term him, William Neal, 
was the first to rectify a curve of any sort ; this curve was the 
semi-cubical parabola ; these rules, of such use in the art of ship- 
building, were first given by him, but as is usual, claimed by Eng- 
lish pretenders. 

The foregoing rules, when applied to the measurement of the 



SHIP-BUILDING AXD NAVAL ARCHITECTURE. 400 

immersed portion of a floating body, as the displacement of a ship, 
are used as follows. 

The ship is considered as being divided longitudinally by equi- 
distant athwartship or transverse vertical planes, the boundaries 
of which planes give the external form of the vessel at the respec- 
tive stations, and therefore the comparative forms of any inter- 
mediate portion of it. 




F 

If the ship be immersed to the line AB, considered as the line 
of the proposed deepest immersion or lading, the curves HLO and 
KMF would give the external form of the ship at the positions Gr 
and I in that line ; and the areas GrHLO, IKMF contained under 
the curves HLO, KMF, the right lines G-H, IK, (the half-breadths 
of the plane of proposed flotation AB at the points Gr and I,) and 
the right lines GO, IF, the immersed depths of the body at those 
points are the areas to be measured ; and if the areas obtained be 
represented by linear measurements, and are set ofi" on lines drawn 
at right angles to the line AB at their respective stations, a curve 
bounding the representative areas would be formed, and the mea- 
surement by the rules of the area contained under this curve, and 
the right line, AB, or length of the ship on the load-water line, 
would give the sum of the areas thus represented, and thence the 
solid contents of the immersed portion of the ship in cubic feet of 
space. In accordance with this application of those rules to mea- 
sure the displacement of the ship, the usual practice is to divide 
the ship into equidistant vertical and longitudinal planes, the lon- 
gitudinal planes being parallel to the load-water section or hori- 
zontal section formed by the proposed deepest immersion. 

To measure the areas of these planes after they have been de- 
lineated by the draughtsman, the constructor divides the depth of 
each of the vertical sections, or the length of each horizontal sec- 
tion, into such a number of equal divisions as will make either one 
or the other of the rules 1 or 2 applicable. If the first rule be 
preferred, the equal divisions must be of an even number, so that 
there may be an odd number of ordinates ; while the use of the 
second rule, to measure the area, will require the equal divisions 
of the base to be in number a multiple of the numeral 3, which 
will make the ordinates to be in number a multiple of the numeral 
3, with one added. From the points of equal divisions in the re- 
spective sections thus determined, perpendicular ordinates are 
drawn to meet the curve, or the external form of the transverse 
planes of the body; and a table for the ordinate3»thus obtained, 
having been made, as shown page 467, the measures by scale of the 
respective ordinates are therein inserted. 



456 THE PRACTICAL MODEL CALCULATOR. 

For the area IKMF, the linear measurements of IK, 1*1, 2*2, 
3-8, 4'4, are taken by a scale of parts, and inserted in the column 
marked 5, page 467, the whole length AB of the load-water line 
being divided into 10 equal divisions, and the area IKMF being 
supposed as the fifth from B, the fore extreme of the load-water 
line. To apply the first rule to the measurement of the area of No. 
5 section, the ordinates are extracted from the table, page 467, and 
operated upon as directed by the rule ; viz. 

Area = { A + 4 P + 2 Q} x ^. 

IK, or first, 1-1, or 2d, 2-2 or 3d, 

4-4, or last, 3-3, or 4th, X 2. 

added together or 2 Q. 

added together = A. and X 4 = 4 P. 

By rule, area = {A + 4 P + 2 Q} x g. 

Whence area = {(IK + 4-4) -f (1-1 -f 3-3) 4 -f 2-2 x 2} x g = 

area IKMF ; and, in a similar manner, may the several areas of 
the other transverse sections be determined. 

When these areas have all been thus measured, they are to be 
summed by the same rules ; the areas themselves being considered 
as lines, and the result will give the solid for displacement in cubic 
feet. To shorten this tedious application of the formula, the ar- 
rangement of having double-columned tables of ordinates was in- 
troduced, as shown on page 484, and for the more ready use of this 
enlarged table, the modifications in the formula 467, before alluded 
to, were adopted, that of 

Area = |A + 4P-f2Q|xg = J2+2P + Q| X j, 

and that of 

Area= |a + 2P4-3qIx-^ = i ^ -hV + 1'5q\ xfr, 

as rendering the required number of figures much less, whereby 
accuracy of calculation is insured and time is saved. 

In using a table of ordinates constructed for this method of cal- 
culation, the linear measurement of the several ordinates of vertical 
section 5 and the corresponding ones of all the others would be in- 
serted in the double columns prepared for them, in the following 
order : — 

In the first and last lines of the enlarged table for the ordinates, 

distinguishable hy -q, in the left-hand column of each pair, the 

measurements of the first and last ordinates of the respective areas 
are placed, and in the right-hand column of each pair one-half of 
such measurements, as being one-half of the first and last ordinates 
of each vertical section or area. In the lines distinguished by 2 P, 
in the left-hand column, the measurements of the even ordinates 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 457 

of each respective area are placed, which having been multiplied 
by two, the result is placed in the respective right-hand columns 
prepared for each vertical section ; while in those lines of the table 
distinguished by Q, the measurements of the ordinates themselves 
are placed in the right-hand columns, as not requiring by the modifi- 
cation of the rules any operation to be used on them, before being 
taken into the sum forming the sub-multiple of the respective areas. 

It may here with propriety be suggested, that in practice the 
insertion of the linear measurements of the ordinates in the table 
in red ink will be found useful, and that after such has been done, 
by the upper line of figures in the table of ordinates thus arranged, 
being divided by two, the second line of figures being multipled 
by two, and so on with the others as shown by the table, and the 
results thus obtained being inserted in their respective right-hand 
columns as before described, great facility and despatch of calcu- 
lation are afforded to the constructor. 

That this method will yield a correct measurement of the areas 
will be evident by an inspection of the terms of the general expres- 

r A ^ 2 7* 

sion of area = <-q- + 2P+Q> X -q-, which are placed against 

the several lines of the table of ordinates. And it will be equally 
apparent, that the sum total of the figures inserted in the right- 
hand columns appropriated to each section is a sub-multiple of 
the area of each section, and that these results arising from the 

use of the form for area of<-o4-2P-fQ> will be one-half of 

those that would be obtained by abstracting the ordinates from the 

table, page 467, and using them in the expression A + 4 P + 2 Q ; 

and therefore to complete the calculation for the areas by the rule, 

2r 
the first results for the areas must be multiplied by -o-, and the 

T , 

last by q, where r is equal to the common interval or equal divi- 
sion of the base in linear feet ; or the part of the expression for 

r A 1 2 r 

areas of<-2+2P-fQl must be multiplied by -q-? to make it 

T 

equivalent to {A -f 4P -f 2 Q} X g. 

The sub-multiples of the areas of the vertical sections thus deter- 
mined, require to be summed together for the solid of displacement, 
and by considering the sub-multiples of the areas to be, as before 
stated, represented by lines or proportionate ordinates, O'Neill's 
rules, by the same table of ordinates with an additional column, may 
be made available to the development of the solid of displacement. 
For the sectional areas being represented by lines, by the first rule, 
one-half the first and last areas, added to the sum of the products 
arising from multiplying the even ordinates or representative areas 
by two, together with the odd ordinates or the areas as given by 



458 THE PRACTICAL MODEL CALCULATOR. 

the tables, and these being placed in the additional column of the 
table prepared for them, the sub-multiple of the solid of displace- 
ment will be given. 

The operation will stand thus : Sub-multiple of each of the 

areas = ^-o+2P + Ql,or each area will be -^ less than the 

full result, and the representative lines for the areas will be dimi- 
nished in that proportion; and having used these sub-multiples of 
the areas thus diminished in the second operation for obtaining the 
sub-multiple of the solid of displacement under the same rule, the 

results will again be -o" less than the true result ; therefore the 

sum thus determined will have to be multiplied bj the quantity 

"o" X ~D~j to give the solid required. In this expression, of 

2r 2/ 

-Q- X -o-, r = the equal distances taken in the vertical planes to 

obtain the respective vertical areas ; r'.= the equal distances at which 
the vertical areas are apart on the longitudinal plane of the ship. 

The displacement being thus determined, by an arrangement of 
an enlarged table of ordinates, the functions arising from the sub- 
multiples of the areas of the vertical sections being placed in O'Neill's 
rules to ascertain the displacement, may be used in the table of 
ordinates to find the distance of the centre of gravity of the im- 
mersed body from any assumed vertical plane ; and also the dis- 
tance that the same point — " the centre of gravity of displacement" 
— is in depth from the load-water or line of deepest immersion, and 
that from the considerations which follow : — 

In a system of bodies, the centre of gravity of it is found by 
multiplying the magnitude or density of each body by its respective 
distance from the beginning of the system, and dividing the sum 
of such products by the sum of the magnitudes or densities. The 
displacement of a ship may be considered as made up of a suc- 
cession of vertical immersed areas ; and if it be assumed that the 
moments arising from multiplying the area of each section by its 
relative distance from an initial plane may be represented by suc- 
cessive lineal measurements, the general rules will furnish the sum- 
mation of such moments ; and the displacement or sum of the areas 
has been obtained by a similar process, from whence, by the rule 
for finding the centre of gravity of a system as before given, the 
distance of the common centre of gravity from the assumed initial 
plane would be ascertained, by dividing the sum of the moments of 
the areas by the sum of the areas or the solid of displacement. 

To extend this reasoning to the enlarged table of ordinates used 
for the second method of calculation : The sub-multiples of the 
respective areas, when put into the formulas to obtain the propor- 
tionate solid of displacement, are relatively changed in value to 
give that solid, and consequently the moments of such functions of 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 459 

the vertical areas will be to each other in the same ratio ; and the 
sum of these proportionate moments, if considered as lines, can be 
ascertained by multiplying the functions of the areas by their rela- 
tive distances from the assumed initial plane, or by the number of 
the equal intervals of division they are respectively from it, and 
afterwards, by the rules, summing these results, forming the sum 
of the moments of the sub-multiples of the functions of the verti- 
cal areas : and the proportionate sub-multiple for the displacement 
is shown on the table ; the division therefore of the former, or the 
sum of the proportional moments of the functions of the areas, by 
the proportionate sub-multiple for the displacement, will give the 
distance (in intervals of equal division) that the centre of gravity 
of the displacement is from the initial plane, which being multi- 
plied by the value in feet of the equal intervals between the areas, 
will give the distance in feet from the assumed initial plane, or from 
the extremity of the base line of the proportional sectional areas 
for displacement. This reasoning will apply equally to finding the 
position of the centre of gravity of the body immersed, both as 
respects length and depth, and on the enlarged tables for construc- 
tion given, (pages 484 and 485,) the constructor, by adopting this 
arrangement, will at once have under his observation the calcula- 
tions o?i, and the results o/, the most important elements of a naval 
construction. 

The foregoing tabular system, for the application of O'Neill's 
rules to the calculations required on the immersed volume of a 
ship's bottom, led to a lineal delineation of the numerical results 
of the tables, and thence the development of a curve of sectional 
areas, on a base equivalent to the length of the immersed portion 
of the body, or of the length at the load-water line. To effect 
this, the sub-multiples of the sectional areas, taken from the tables 
for calculation, are severally divided by such a constant number as 
to make their delineation convenient ; then these thus further 
reduced sub-multiples of the areas, being set off at their respective 
positions on the base, formed by the length of the load-water line, 
a curve passed through the extreme points of these measurements, 
will bound an area, that to the depth used for the common divisor 
would form a zone, representative of the solid of displacement. 
The accuracy of such a representation will be easily admitted, if 
the former reasoning is referred to. 

To obtain the solid of displacement from this representative area, 
the load-water line or plane of deepest immersion is considered as 
being divided lengthwise into two equal parts, which assumption 
divides the base of the curve of sectional areas also into two equal 
portions : the line of representative area to that medial point is 
then drawn to the curve, and triangles are formed on each side of 
it by joining the point where it meets the curve with the extremi- 
ties of the base line ; this arrangement divides the representative 
area into four parts, two triangles which are equal, viz. 1 and 2, 
and two other areas which are contained under the hypothenuse of 



460 THE PRACTICAL MODEL CALCULATOR. 

these triangles and the curves of sections, or 3 and 4 of the an- 
nexed diagram. 

Diagram of a Curve of Sectional Areas. 




"5" 

ABCDA equal sectional area, representative of the half displace- 
ment as a zone of a given common depth. 

AC equal the length of the load-water section from the fore-side 
of the rabbet of the stem to the aft-side of the rabbet of the post, 
and D the point of equal division. 

BD, the representative area of half the immersed vertical sec- 
tion at the medial point D, joining B with the points A and C, will 
complete the division of the representative area ABCDA. 

ABD and CBD, under such considerations, are equal triangles. 

BECB, BFAB, areas, bounded respectively by the hypothenuse 
AB or BC of the triangles and the curve of sectional areas ; and, 
supposing the curves AFB and BEC to be portions of common 
parabolas, the solid of displacement will be in the following terms : 

The area of each of the triangles is equal to \ of AC X BD ; 
hence the sum of the two = J of AC x BD ; the hypothenuse AB 

or BC = J[(^) + BD^], and the area of BECB if consi- 

I AC 2 
dered as approximating to a common parabola = J I ( ~o~) + BD- J 

X f of the greatest perpendicular on the hypothenuse BC. 
Area of BFAB under the same assumption = J T \~9~) "'" BD- J 

X f of the greatest perpendicular on the hypothenuse AB ; whence 
the whole displacement will be expressed by J AC X BD x 

j\\~o~j + BD^J X f of the greatest perpendicular on the hypo- 
thenuse BC+/r(-2-) + BDM X f of the greatest perpendi- 
cular on the hypothenuse AB. 

By a similar method, from the light draught of water, or the 
depth of immersion on launching the ship, the light displacement, 
or the weight of the hull or fabric, may be delineated and esti- 
mated ; and the representative curve for it being placed relatively on 
the same base as that used for the representative curve for the load 
displacement, the area contained between the curve bounding the 
representative area for the load displacement, and the curve bound- 
ing the representative area for the light displacement, will be a repre- 
sentative area of the sum of the weights to be received on board, 
and point out their position to bring the ship from the light line 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 



461 



of flotation, or the line of immersion due to the weight of the hull 
when completed in every respect, to that of the deepest immersion, 
or the proposed load-water line of the constructor — a representa- 
tion that would enable the constructor to apportion the weights to 
be placed on board to the upward pressure of the water, and thence 
approximate to the stowage that would insure the easiest movements 
of a ship in a sea. 

By an inspection of the diagram of the curve of sectional areas, 
it will clearly be seen that the representative area for displacement 
under the division of it, into the triangles 1 and 2, and parabolic 
portions of the area 3 and 4, will point out the relative capacities 
of the displacement, under the fore and after half-lengths of the 
base or load-water line ; for, by construction, the triangles ABD 
and CBD are equal, and therefore the comparative values of the 

areas BECB and BFAB, or of J[_{-^) + BD^] x f of the 
greatest perpendicular on the hypothenuse BO, compared with 
J r (—^ J + BD^ J X f of the greatest perpendicular on the hypo- 
thenuse AB, or of the relative values of the greatest perpendicu- 
lars on the hypothenuses BC and AB, will give the relative capaci- 
ties of the fore and after portions of the immersed body or the dis- 
placement. 

The representative area ABCDA admits also of a measurement 
by the second rule. 

Let BD, as before, be the representative area at the middle point. 




A6 5 4 D 3 2 IC 

Divide AD or DC into three equal portions, then the equal divi- 
sions being a multiple of 3, the second rule is applicable to 
measure the areas ABD A or BCDB ; for the area ABD A = 

I 6,6 -f BD + 2 X + 3 {4,4 + 5,5} \~; 6,6 = ; 
= j BD + 3 {4,4 + 5,5} V-g- ; and area BCDB = 



1 1,1 + BD -I- 2 X + 3 X {2,2 -f 3,3} |-^,wherel,l = 
I BD -f 3 X {2,2 + 3,3} }-g- = BCDB, and the displace- 
ment = | BD + 3 x {4,4 + 5,5} |^-f|BD-f-3x{2,2H-3,3} l 



X -^ X by the constant divisor of the areas, or the depth of the 
zone in feet. 



462 THE PRACTICAL MODEL CALCULATOR. 

The rules given by O'Neill for the measurement of the im- 
mersed portion of the body of a ship, having been theoretically 
stated, the practical application of them will be given on the con- 
struction. 

The immersed part of a ship, being a portion of the parallelopi- 
pedon formed by the three dimensions ; — length on the load-water 
line, from the foreside of the rabbet of the stem to the aftside 
of the rabbet of the stern-post ; extreme breadth in midships of the 
load-water section ; and depth of immersion in midships from the 
lower edge of the rabbet of the keel ; — it would seem that the first 
step towards the reduction of the parallelopipedon, or oblong, into 
the required form, would be to find what portion of it would be of 
the same contents as the proposed displacement of the ship — a 
knowledge of which would enable the constructor, by a comparison 
of the result with a similar element of an approved ship, to deter- 
mine whether the principal dimensions assumed would (under the 
form intended) give an immersed body equal to carrying the pro- 
posed weights or lading. 

The relative capacities of the immersed bodies contained under 
the fore and after lengths of the load-water line must next be fixed, 
and the constructor in this very important element of a construc- 
tion will find little to guide him from the results of past experience 
and practice. From deductions on approved ships of rival con- 
structors it will be developed, that in this essential element, " the 
relative difference between the two bodies," they vary from 1 to 
13 per cent, on the whole displacement. 

The relative capacities of the fore and after bodies of immersion 
under the proposed load-water line would seem at the first glance 
of the subject to be a fixed and determinate quantity, as being a 
conclusion easily arrived at from a knowledge of the proportions 
due to the superincumbent weights — under such a consideration, the 
weight of the anchors, bowsprit, and foremast would necessarily be 
supposed to require an excess in the body immersed under the fore 
half-length of the load-water line over that immersed under the 
after half-length of the same element. 

In a ship, the necessary arrangement of the weights, to preserve 
the proposed relative immersion of the extremes or the intended 
draught of water, would be pointed out by a delineated curve of 
sectional areas, described as before directed ; but a want of that 
system, or of some other, has often caused an error in the actual 
draught of water, and that under a great relative excess of the 
volumes of displacement in the fore and after portions of the im- 
mersed body. 

The men-of-war brigs built to a construction-draught of water 
12 ft. 9 in. forward, 14 ft. 4 in. abaft, giving 1 ft. 7 in. difference, 
had under such a construction a difference of displacement between 
the immersed bodies under the fore and after half-lengths of the 
load-water line that was equivalent to 10*4 tons for every 100 tons 
of the vessel's total displacement or weight ; but these ships, when 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 463 

Stowed and equipped for sea, came to the load-draught of water of 
14 ft. 2 in. forward, 14 ft. 3 in. aft, — difference 1 inch, or an immer- 
sion of the fore extreme of 18 inches more than was intended by 
the constructor. The reason of this practical departure from the 
proposed line of flotation of the constructor was, that the inter- 
nal space or hold of the ship necessarily followed the external form, 
giving a hold proportionate to the displacement contained under the 
several portions of the body ; but an injudicious disposal of the 
stores (in placing the weights too far forward) made them more than 
equivalent to the upward pressure of the water at the respective 
portions of the proposed immersion of the body, and thence arose 
the error or excess in the fore immersion by giving a greater draught 
of water than was designed. The stowage of a ship's hold, under 
a reference to the representative area for the displacement, con- 
tained between the curves of sectional areas developed for the light 
and load displacements, would prevent similar errors under any 
extent to which the relative capacity of the two bodies might be 
carried. This relative capacity of the two bodies will affect the 
form of the vessel's extremes, giving a short or long bow, a clear 
or full run to the rudder ; for the whole displacement being a fixed 
quantity, if the portion of it under the fore half-length of the load- 
water line be increased, it must be followed by a proportionate dimi- 
nution of the portion of the displacement under the after half-length 
of the load-water line, so that the total volume of the displacement 
may remain the same, which arrangement will give a proportionately 
full bow and clean run, and vice versd. 

The curve of sectional areas under the foregoing considerations 
is also applicable to a comparison of the relative qualities of ships 
of the same rate, by showing at one view the distribution of the 
volume of displacement in each ship, under the draught of water 
which has been found on trial to give the greatest velocity ; based 
on which, deductions may be made from the relative capacities of 
the bodies pointed out by the sectional curves, that will serve to 
guide the naval constructor in future constructions. 

The curve of sectional areas is also available for forming a scale 
to measure the amount of displacement of a ship to any assumed 
or given draught of water. To effect this, on the sheer draught or 
longitudinal plan of the ship between the load-water line, or that 
of deepest immersion, and the line denoting the upper edge of the 
rabbet of the keel, draw intermediate lines parallel to . the load- 
water line as denoting lines of intermediate immersion between the 
keel and load-water line ; these lines may be placed equidistant 
from each other,, but they are not necessarily required to be so. 
Find the curve of sectional areas, due to each immersion of the 
ship denoted by these lines, and measure the areas bounded respec- 
tively by these curves, in the manner as before directed for the load 
displacement : these results will give the magnitudes of the im- 
mersed portions of the body in cubic feet, which being divided by 35, 
the mean of the number of cubic feet of salt or fresh water that 



464 THE PRACTICAL MODEL CALCULATOR. 

are equivalent to a ton in weight, will give their respective weights 
in tons. 

Assume a line of scale for depth, or mean draught of water, the 
lower part of which is to be considered the underside of the false 
keel of the ship, and set off on this line, by means of a scale of 
parts, the depths of the immersions at the middle section of the 
longitudinal plan ; draw lines (at the points thus obtained) perpen- 
dicular to this assumed line for depth or draught of water, and 
having determined a scale to denote the tons, set off on each line 
bj this scale the tons ascertained by thef curves of sectional areas 
to be due to the respective immersions of the body; then a curve 
passed through these points will be one on which the weights in tons 
due to the intermediate immersions of the body may be ascertained ; 
or, the displacement of a ship to the mean of a given draught may 
be found by setting up the mean depth on the scale, showing the 
draught of water — transferring that depth to the curve for tonnage, 
and then carrying the point thus obtained on the curve for tonnage 
to the scale of tons, which will give the number of tons of displace- 
ment to that depth of immersion or draught of water. 

Description of the several plans to he delineated hy the draughts- 
man, previous to the commencement of the calculations. 

Sheer Plan. — A projection of the form of the vessel on a longi- 
tudinal and vertical plane, assumed to pass through the middle of 
the ship, and on which the position of any point in her may be 
fixed with respect to height and length. 

Half -breadth Plan. — The form of the vessel projected on to a 
longitudinal and horizontal plane, assumed to pass through the ex- 
treme length of ship, and on which the position of any point in 
the ship may be fixed for length and breadth. 

Body Plan. — The forms of the vertical and athwartship sections 
of the ship, projected on to a vertical and athwartship plane, 
assumed to pass through the largest athwartship and vertical sec- 
tion of her, and on which plan the position of any point in the ship 
may be fixed for height and breadth. 

These plans conjointly will determine every possible point re- 
quired ; for, by inspection, it will be found — 
That the sheer and half-breadth plans have 

one dimension common to both, viz.: Length. 

Half-breadth and body plane Breadth. 

Sheer and body plane Height. 

For sheer plan gives length and height ^ ^^ ^^^ ^^^^ 

Half-breadth plan gives length and breadth > r> • t 

Body plan gives breadth and height J ^ 

Which dimensions form the co-ordinates for any point in the solid, 
and must determine the position of it. 

The point C in the load-water section AB, has for its co-ordi- 
nates to fix its position. 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 465 

The length, 1-5 of the half-breadth plan. 

Height, 5-C of the sheer plan, 

And the breadth, 1*0 of the body plan of section. 

And the same for any other point of the solid or of the ship. 

In the sheer plan, AB represents the line of deepest immersion, 

a a, bb, c c, dd, lines drawn parallel to that line at a distance of 

•9 feet, making with AB an odd number of ordinates for the use of 

r 
the first general rule for the area, where area = {A + 4P -f 2 Q} X q, 

and A = the sum of the first and last ordinates. 

P = the sum of the even ordinates, as 2, 4. 

Q = the sum of the odd ordinates, as 3, &c. 

The line AB, or length of the load-water line, is bisected at C, 
and AC, CB are thence equal; C being the middle point of the 
load-water line, the spaces BC, AC are again divided into four 
equal divisions, giving five ordinates for each space, at a distance 
apart of 5*5 feet. 

This arrangement will give the immersed body of the vessel, as 
being divided into two parts under an equal division of the load- 
water line, and an odd number of ordinates in each section of the 
body for the application of the first general rule given for finding 
the areas of the vertical sections and thence the displacement. 

The half-breadth plan delineates the form of the body immersed 
for length and breadth, the line AB of the sheer plan being repre- 
sented in the half-breadth plan by the line marked AB, and 
a a, bb, c c, dd, of the sheer plan by the lines similarly distin- 
guished in the half-breadth plan. 

The body plan gives the form of the body in the depth, the lines 
distinguished 5'b in the sheer and half-breadth plans being in the 
body plan developed by the curve 5'5-5, giving the external form 
of the ship at the section 5*5 ; the same reasoning applies to the 
other divisions of the load-water line AB. 



A pile of 400 lbs. weight is driven by the last round of 20 blows 
of a 500 lbs. heavy ram, falling from a height of 5 feet; 6 inches 
deeper, what resistance will the ground ofi'er, or what load will the 
pile sustain without penetrating deeper ? ^^ 

.^ Here G = 400, G, = 700 lbs., H = 5, and s = ^^ = 0-025 feet, 

whereby it is supposed that the pile penetrates equally far for 
each blow. 

/ 700 x2 400 X 5 / 7 x2 
P = ( 700 -f 400 ) -0-025 = (n) ^ 8^^^^ =^^400 lbs., 
the ram, not during penetration, remaining upon the pile. 
700^ X 5 4900 

^ ~ 1100 X ~02'' ~ ~lT" ^ ^^^ ~ 89100 lbs., the ram remain- 
ing upon the pile during penetration. 

For duration, with security, such piles are only loaded from j^ 
to ^n of their strength. 

30 



466 



THE PRACTICAL MODEL CALCULATOR. 



o 






•o 


rH « 1 W TK.n^ 


1 


^ 




/ 


y 


«o 




Principal Divieiisions. 

Ft. In. 

Length for Tonnage 45 

Keel for Tonnage 36 lOf 

Breadth for do 13 6 

Biuihen in Tons 35 || 



AB, Load-water Line. 

aa\ 

h h I Lines parallel to AB at the 

cc \ distance of '92 feet apart. 



dd) 



Draught of Water. 



Ft. In 

Afore 4 G 

Abaft 7 6 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 



467 




fe. 



Calculations required for the construction draw- 
ing of a yacht of 36 tons. — 1st, Usual mode of 
calculating the displacement hy vertical and 
horizontal sections. 

Table of Ordinates for Yacht of 36 Tons. 



Distinguish--) 

ing No. of y 
the sections. J 

I'A 
2'P 
3'Q 
4'P 

5' A 


1 

•35 
•3 


2 

3^ 
24 
hi 
VO 
•4 


3 

5^ 
4^ 
3^ 
?2 
VB 


4 (5) 6 

6-0 6-3 6-1 
5-6 5-6 5-5 
44 5^4^ 
?2 3^34 
2^124 2^ 


7 

5-4 
44 
34 
24 
14 


8 

2-6 
1-7 
1-1 


9 


r = the distance be- 
tween the ordi- 
nates used for 
the vertical sec- 
tion = -92 feet. 

r'— the distance be- 
tween the ordi- 
nates used for 
the horizontal 
sections = 5-5 
feet. 



From this Table the following application of 
O'Neill's rule, No. 1, is usually made to obtain 
the volume of displacement to the draught of 
water shown on the drawing as the load-water 
line, or line of proposed deepest immersion, de- 
signated by AB. 

General terms of the rule : — 



Area 



A + 4P + 2Q 



3- 



To find ^ the area of vertical section 1, fore 
body : — 



A=sum of 
the first 
and last 



•41 



4P=fourtimesthesum^ '35 
of the even ordinates, > 
or of (2) and (4) J -25 



•6 = A 



•60=P 
4- 

2-4 =4 P 

2 Q = twice the sum of the odd \ -3 = Q 
ordinates, or of (3) j x 2 

"^ = 2Q 
Whence the area, which is equal to 

|a + 4P + 2q\x^=|-6 + 2-4 4-6|x^. 



•92 
3-6 X -g- = 1-2 X -92 = 1-104 

section 1. 



J area of 



Which sum is half the area of the section 1, and is kept in that 
form of the half-measurement for the convenience of calcula- 
tion. 



468 THE PRACTICAL MODEL CALCULATOR. 

Fore Body. 
Vertical Section 2. 

3-0 2-4 1-7 

j4 1;0 __2 

3-4 = A 3-4 ^ P 3-4 = 2 Q 

4 

iM = 4P 
3-4 = A 
3-4 = 2 Q 
20-4 = A + 4P + 2Q 
'•92 = r 
"408' 
1836 



3) 18-768 

6-256 = J area of Section 2. 

Vertical Section 3. 
5-0 4-2 3-2 

rs 2^ _2 

6-3 = A 6-4 = P 6-4 = 2 

4 
25-6 = 4P 
6-3 = A 
6-4 = 2 Q 
38-3 = A + 4P + 2Q 



•92 



r 



766 
3447 

3)35-236 



11-745 = A-f4P + 2QXg = i area of Section 3 

Vertical Section 4. 

6-0 5-6 4-4 

2^0 3^ _2 

8^ = A 8-8 = P 8-8 = 2Q 



35-2 = 4 P 

8-0 = A 

8-8 = 2 Q 
52-0 = A + 4P + 2Q 

-92 = r 



1040 
4680 



3)47-840 



15-946 =A+4P-f2QXg = } area of Section 4. 



i 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 469 

Vertical Section 5. 
6-3 5-6 5-0 

8-7 = A 9-4 = P Wo == 2 Q 

4 
37 -6 = 4 P 
8-7 = A 
lO'Q = 2Q 

56-8 = A-f4P + 2Q 
•92 =r 
» 1126 

5067 
3) 51-796 

17-265 = A -f 4P + 2Q x | = J area of Section 5. 

Half areas of Vertical Sections 1, 2, 3, 4, and 5. 

No. 1 1-104 feet. 

2 6-256 

3 11-745 

%- 4 15-946 

5 17-265 

Displacement of the body under the fore half-length of the load- 
water line by the vertical sections, or the summation of the vertical 
areas 1, 2, 3, 4, and 5, by the formula for the solid, as being 
equal to 

«^ A' 4- 4 P' + 2 Q' I X o- where A' = sum of 1st and 5th areas. 

P' = " 2d and 4th areas. 
Q' = " 3d area. 
And r' = distance between the vertical sections, or 5-5 feet. 
1... 1-104 2... 6-256 3. ..11-745 = Q' 

5.. .17-265 4. .. 15-946 2 

18-369 = A' 2M02 = F 23-490 = 2 Q' 

4 

88-808 = 4P' 
18-369 = A' 
23-490 = 2Q' 
130-667 = A' -f 4F -f 2Q' 



5-5 = 



653335 
653335 

3) 718-6685 , 

239-556 = AM^F+W x ^=cubic ft. of 

2 space in 1^ fore-body. 
479-112 = cubic feet of space in fore-body. 



470 THE PRACTICAL MODEL CALCULATOR. 

Displacement of the body immersed under the after half-length 
of the load-water line by the vertical areas 5, 6, 7, 8, and 9 of the 
Table of ordinates. 

Vertical Section 6. 
5, as fore body. 6-1 5-5 4*6 = Q 

17-265 2^ _34 _2 

8-1 = A 8-9 = P 9-2 = 2 Q 

4 . 
35^ = 4P 
8-1 = A 
9-2 = 2 Q 
52-9 = A + 4P4-2Q 

♦92 = r 
1058 
4761 

3) 4"M68 

. 16-222 = A+4P4-2Qx^={ I -^^^/l 

Vertical Section 7. 
5-4 4-4 3-4 = Q 

14 24 _2 r 

6-8 = A M = P 6-8 = 2 Q 

4 

2^2 = 4 P 

6-8 = 2 Q 

6-8 = A 
40-8 = A + 4P-h2Q 

-92 = r 

816 
3672 



3) 37-536 

12-512 = A+4P-f2Qx3={|-^V7. 
Vertical Section 8. 
3-7 2-6 1-7 = Q 

j6 21 j2 

4-3 = A 3-7 = P 3-4 = 2Q 

4 
14-8 = 4 P 
4-3 = A 
3-4 = 2 Q 
22-5 = A + 4P + 2Q 
•92 = r 
450 
2025 



3)20-700 ^ .^ . 

W- = AT4PT2Qx|= {!-//. 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 471 

Vertical Section 9. 
•4 -35 . -3 = Q 

^ _^ _2 

•6 = A -60 = P "^ = 2 Q 

4 

2-4 = 4 P 

•6 = A 
j6 = 2Q 

3-6 = A +4P -f 2Q 
•92 = r 
72 
324 

3)3-312 ^ 

1-104 = A + 4P4-2QXg==} area of Section 9. 

Half areas of the vertical sections 5, 6, 7, 8, and 9. 

Sections. Areas. 

5 17-265 

6.. 16-22 

7 12-512 

8 6-9 

9 1-104 

Displacement of the after-body under the after half-length of the 
load-water line by the vertical sections, or the summation of the 
immersed areas of the vertical sections 5, 6, 7, 8, and 9 by the 
formula for the solid as being equal to 

A' + 4 P' + 2 Q' X ^ 

where A' = sum of the 5th and 9th areas. 
P' = " 6th and 8th areas. 

Q' = " 7th area, 

and r' = the distance between the vertical sections, or 5-5 ft. 

5. ..17-265 6. ..16-22 7. ..12-512 = Q' 

9... 1-104 8... 6-900 2 

18-369 = A' 23-120 = F 25-024 = 2 Q' 

4 

92-480 = 4 F 
25-024 = 2 Q' 
18-369 = A' 
135-873 = A' + 4 F + 2 Q' 

5;5 = r 

679-365 
67-936 

3 ) 747-3015 , 

249-1005 =A'-f 4F+2Q'x^=cubicft. 

2 in J after-body. ^ 

498-2010 After-body in cubic ft. of space. 



472 THE PRACTICAL MODEL CALCULATOR. 





Displacement of Forc-bocly by Horizontal Sections. 






Horizontal /Section V. 




0-4 




6-0 


5-0 = Q 


6-3 




3-0 


2 


6-7 = 


= A' 


9-0 = P 

4 

36-00 = 4P 

10-00 = 2 Q 

6-70 = A 

52-70 = A + 4P -f-2Q 
5-5 = r 


10-0 = Q 




2635 








2635 






K 


289-85 






96-61 = A + 4P + 2Qx^ = 

o 


= J area of Sec 






Horizontal Section 2'. 




•35 




5-7 


4-2 = Q 


5-60 




2-4 ^ 


2 


5-95 


= A 


8-1 = P 

4 
324 = 4P 
8-4 = 2 Q 
5-95 = A 
46-75 = A + 4P + 2Q 

5-5 = r 
23375 
23375 


8-4 = 2 C 




3 


) 257-125 






85-708 = A + 4P + 2Qxr = 

o 


= J area of Sec 






Horizontal Section 3'. 




•3 




4-4 


3-2 = Q 


5-0 




1-7 


2 


5-3 = 


= A 


6-1 = P 

4 
24-4 = 4 P 

5-3 = A 

6-4 = 2 Q 
36-1 = A4-4P + 2Q 

5-5 = r 


6-4 = 2 Q 



1805 
1805 
3U98-55 



66-18 = A + 4P + 2Qx^=i area of Section 3'. 
o 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 473 







Horizontal Section 4'. 




•25 




3-2 


2-2 = Q 


3-8 




1-0 


2 


4-05 = 


A 


4-2 =P 
4 

16-8 =4P 


4-4 = 2 Q 


» 




4-05 = A 
4-40 = 2 Q 

25-25 = A + 4P + 2Q 
5-5 = r 





12625 
12625 

3 ) 138-875 







46-291 = A + 4P -f 2Q X 


r 
3 


_ j 1 area of 
~ \ Section 4' 






Horizontal Section 5'. 






•2 

2-4 




2-0 
-4 




1-3 = Q 

2 


2-6 = 


A 


2-4 = P 
4 

9-6 = 4P 

2-6 = A 
2-6 = 2 Q 

14-8 = A-f4P-f2Q 
5-5 = r 




2-6 = 2 Q 



740 

740 

3 ) 81-40 

27-13 = A + 4P + 2Q x | = { |^ ^^ «/, 

Displacement of the fore-body under the fore half-length of the 
load-water line by horizontal sections, or the summation of the 
horizontal sections of the fore-body 1', 2', 3', 4', and 5', by tlie 
formula for the solid, as being equal to 

A' + 4F-f 2Q' x^; 

where A' = sum of the I'st and 5'th areas ; 

P' = " 2'd and 4'th areas; 

Q' = " 3M area ; 

and r = the distance between the horizontal sections, or -92 feet. 
Half areas of the Horizontal Sections 1', 2', 3', 4^ and 5'. 



1' = 96-61. 
2' = 85-708. 
3' = 66-18. 



4' = 46-29. 
5' = 27-13. 



474 THE PRACTICAL MODEL CALCULATOR. 



Areas. 

r.. .96-61 
5'. ..27-13 


Areas. 

2'. ..85-708 
4'. ..46-290 


Areas. 

3'.. .66-18 = Q' 

2 


123-74 = A' 


131-998 = F 
4 


132-36 = 2 Q' 




527-992 = 4F 
123-740 = A' 
132-360 = 2 Q' 






784-092 = A' + 4F 
-92 = r 


+ 2Q' 



1568184 
7056828 



3)721-36464 

cubic ft. in 
rfore-body. 



240-45 =A'+4F-{-2Q'x|=|j' 



480-90 = fore-body by horizontal sections in 
cubic feet of space. 

Displacement, by horizontal sections of the body immersed under 
the after half-length of the load-water line, or by the horizontal 
areas 1', 2', 3', 4', and 5', of the table of ordinates. 

Calculated areas of 1', 2,' 3', 4', and 5'. 

Section V After-body, 

5-4 = Q 
2 

10-8 = 2 Q 



6-3 
•4 




6-1 
3-7 




6-7 = 


= A 


9-8 = 
4 


P 






39-2 = 

10-8 = 

6-7 = 


4P 
:2Q 
A 






56-7 = 
5-5 = 


A + 4P + 2Q 
r 






2835 
2835 






3)311-85 





103-95 = A + 4P + 2Qx^={g*--»^_ 



I 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 475 

Section 2' After-body, 



fy-^ 5-5 
•35 2-6 
5-95 = A 8-1 = P 
4 


4-4 

2 
8-8 = 2 Q 


32-40 = 4P 
5-95 = A 
8-80 = 2 Q 




47-15 = A + 4P + 2Q 

5-5 = r 
23575 
23575 




3)259-325 ^^ 




86-441 = A + 4P + 2Qx^ = 


= J area of Section 2'. 


Section W After-body, 
5-0 4-6 
•3 1-7 

5-3 = A 6-3 = P 
4 


• 

3-4 = Q 

•2 

6-8 = 2Q 


.25-2 = 4P 
5-3 = A 
6-8 = 2 Q 




37-3 = A + 4P + 2Q 
5-5 = r 





1865 
1865 



3) 205-15 ^, 

68-38 = A + 4P+2QXg=J area of Section 3'. 

Section 4' After-body. 



3-8 3-4 
•25 1-1 
4-05 = A 4-5 = P 
4 


2-4 = Q 

2 
4-8 = 2 Q 


18-00 = 4 P 
4-05 = A 

4-80 = 2 Q 

26-85 = A + 4P + 2Q 
5-5 = r' 





13425 
13425 
3)147-675 



49-225 = A-f4P + 2QXg- = J area of Section 4'. 



4*76 THE PRACTICAL MODEL CALCULATOR. 

Section 5' After-hody. 
2-4 2-0 1-4 = Q 

j2 ^ 2 

2-6 = A 2-6 = P M=2Q 

4 
10-4 == 4P 
2-8 = 2 Q 
2-6 = A 
15-8 = A + 4P + 2Q 
y 5-5 = r' 
790 
790 

3) 8MQ ^, 

28-96 = A + 4P + 2QXg- = } area of Section 5'. 

Displacement by •horizontal sections of the afl^r-body under the 
after half-length of the load-water line, or the summation of the 
horizontal sections of the after-body, 1', 2^, 3', 4', and 5', by the 
formula of the solid, as being equal to 

A' -f 4 F -f 2 Q' X ^. 
Half areas of the After Horizontal Sections, V, 2\ 3', 4', and 5'. 

Sections. Areas. 

V 103-95. 

2' 86-44. 

3' 68-38. 

4' 49-22. 

5' 28-96. 

Areas. Areas. Areas. 

1'.. .103-95 2^. .86-44 3^.. 68-38 = Q' 

5'... 28-96 4'. .^49^22 2 

132-91 = A' 135-66 = F 136-76 = 2 Q' 

4 

542^ = 4F 

132-91 = A' 

136-76 = 2Q' 

812-31 = A'*+ 4F + 2Q' 

'^2 = r 

162462 
731079 

3) 747-3252 ^ 

249-1084 = A'-f 4P'-f 2Q'x 3 =cubicft.of 
2 J after-body by horizontal sections. 
498-2168 = After-body by horizontal sections 
in cubic feet of space. 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 477 

DISPLACEMENT. 

By Vertical Sections. By Horizontal Sections. 

Cubic Feet. Cubic Feet. 

Fore-body (p. 469) 479-11 Fore-body (p. 474) 480-900 
After-body (p. 471) 498-20 After-body (p. 476) 498-216 

Sum 977-30 Sum 979-116 

Half 488-65 Half 489-558 

Cvibic Feet. 

By Horizontal Sections 979-116 

By Vertical Sections .977-300 

Difference... 1-816 cubic feet. 

Cubic Feet. 

979-49 = capacity or displacement in cubic feet of space. 

The mean weight of salt and fresh water gives 35 cubic feet of 
space, when filled with water, to be equivalent to a ton avoirdupois; 
thence the displacement in cubic feet of space being divided by 35 
will give the weight of the volume displaced in tons avoirdupois ; 
or 979-49 being divided by 35 gives 

5) 979-49 

7) 195-898 

27*985 tons, the weight of the calculated 
immersed body in tons. 

AREA or THE MIDSHIP SECTION, OR OF THE GREATEST TRANSVERSE 

SECTION. 

Section at 5. 
1-1. ..6-3 2-2. ..6-0 3-3. ..4-8 = Q 

5-5. ..j2 4-4. ..2;3 _2 

6-5 = A 8-3 = P 9-6 = 2 Q 
4 

33-2 = 4 P 
6-5 = A 

9-6 = 2 Q 

49-3 = A + 4P-f2Q 

r 
1-25 = 7. where r = the depth, from 1 to 5, di- 

986 ^ ^^ *^* 

493 

3)61-625 

20-541 = A 4- 4P -f 2Q x J = / 2, ^^^^ ^^ °^^^- 
2 ^^ I ship section. 

41-082 == Area of midship section without keel. 



478 



THE PRACTICAL MODEL CALCULATOR. 



LOAD-WATER LINE. 

Area of the load-water line, or area of the assumed deepest plane 
of immersion, delineated on the half-breadth plan, and marked by 
the curve AB. From the table of ordinates, p. 467, we have — 



•4 3-0 
•4 6-0 

•8 - A ^-1 
^ - ^ 3-7 

18-8 - P 
4 

75-2 = 4 P 

•8 = A 
33-4 = 2 Q 


5-0 
6-3 
5-4 

16-7 = Q 
2 

33-4 = 2 Q 


109-4 = A + 4P + 2Q 
5-5 = r' 




5470 
5470 




3)601-70 


/v.r .- , „ , , 


200-56 = A + 4P + 2Q xl-=J t area ot loaa- 

o [ water line. 
200*56 = } area of load-water section in superficial feet. 

2 



401-12 = area of load-water section, which amount of area being 
divided by 12, will give the number of cubic feet of space that would 
be contained in a zone of that area of an inch in depth, and that 
result being again divided by 35, as the number of cubic feet of 
water equivalent to a ton in weight, will give the number of tons 
that will immerse the vessel one inch at that line of immersion. 
12)401-12 = area of load-water section in superficial feet. 
5 ) 33*42 = cubic feet in zone of one inch in depth. 
7 )6-684 

•955 = tons to the inch of immersion at load-water line. 



CENTRE or GRAVITY OF THE DISPLACEMENT. 

Estimated from Section 1, considered as the Initial Plane. 



Distinguishing 
No. of the Areas. 



)4 Vertical 
Areas. 



1 1-104 X 000*000 

2 6*256 X 1 6*256 

3 11*745 X 2 23*490 

4 16*069 X 3 48*207 

5 17*265 X 4 69*060 

6 16*222 X 5 81*110 

7 12*512 X 6 75-072 

8 6*900 X 7 48*300 

9 1*104 X 8 8*832 



SHIP-BTTILDIXG AND NAVAL ARCHITECTURE. 



479 



Moments placed in the Rule. 

r' 

Sum=A + 4P + 2Qx-o 



000-000 

8-832 

8-832 = A 



6-256 
48-207 
81-110 

48-300 

,183-873 = P 
4 



23-490 
69-060 
75-072 

167-622 = Q 

2 



335-244 = 2 Q 



735-492 = 4 P 

8-832 = A 
335-244 = 2Q 

1079-568 =A + 4P + 2Q 
5;5 = r' 

5397840 
5397840 

3) 5937-6240 

1979-208 



= A + 4P + 2Qxl- = 
o 

sum of the moments of half the displacement from section 1, in in- 
tervals of space of 5-5 ft. ; and the half displacement in cubic feet 
by vertical sections is 488-650 (p. 477) cubic ft. ; whence it is 
found, by dividing the moment 1979-208 by 488-650, that the dis- 
tance of the centre of gravity of displacement from the section 1 
is as follows : — 

488-65 ) 1979-208 ( 4-05 intervals from 1. 
195460 interval = 5-5 ft. 

• 246080 

244325 

1755 therefore 4-05 X 5-5 = 22-27 ft. = 
distance of the centre of gravity 
of the calculated immersed body 
from 1. 



DEPTH OF THE CENTRE OF GRAVITY OF THE DISPLACEMENT BELOW 
THE LOAD-WATER SECTION. 



Fore-body. After-body. 



ictioa 


6 

r 


Areas. 

r 96-61 
85-708 
66-18 
46-29 
27-13 




Areas. 

fl03-95 


Sum of the Areas. Moments. 

200-56 X = 000-000 


9/ 


86-44 


172-148 X 1 == 172-148 


8' 


68-38 


134-56 X 2 = 269-12 


4' 


49-22 


95-51 X 3 = 286-53 


5' 


28-96 


56-09 X 4 = 224-36 



480 



THE PRACTICAL MODEL CALCULATOR. 



000-00 
224-36 


A 


172-148 

286-530 

458-678 = P 
4 


269-12 = Q 

2 


224-36 = 


538-24 = 2 Q 






1834-712 = 4 P 
224-360 = A 
538-240 = 2 Q 








2597-312 = A + 4P + 2Q 
•92 = ?• 



5194624 

23375808 

3)2389-52704 



796-509 =A + 4P + 2Qx^ = 



sum of the moments of the half displacement calculated from the 
load-water line : the half displacement by horizontal sections is 
489-588 (p. 477) cubic feet ; the sum of the moments of the half 
displacement 796-509 ft., being divided by that quantity, will give 
the distance in intervals of -92 ft. ; the centre of gravity of dis- 
placement is below the load-water line. 



Half solid of 
displacemeat. 



489-558 ) 796-509 ) 1-62 intervals of -92 feet ; therefore 



489558 

3069510 

2937348 

1321620 
979116 

342504 



1-62 
X -92 

324 
1458 

1-4904 ft. = the distance the cen- 
tre of gravity of the calcu- 
lated immersed body is be- 
low the load-water section. 



DISTANCE OF THE CENTRE OF GRAVITY OF THE AREA OF THE LOAD- 
WATER SECTION FROM SECTION 1. 



No. of Section. 


Ordinates of Section 1 
from the Table, p. 467. 


Distances of them in 

intervals of 5-5 ft. 

from Section 1. 


Moments : being the Pro- | 

dnct of the Areas by the 

respective Distances. 


1 

2 
3 
4 
5 
6 
7 
8 
9 


•4 
3-0 
5 
6-0 
6-3 
61 
5-4 
3-7 

•4 




1 

2 
3 
4 

5 
6 

7 
8 


000-00 
3-0 
100 
18-0 
25-2 
30-5 
32-4 
25-9 
3-2 



SniP-BUILDING AND NAVAL ARCHITECTURE. 481 

The moments, for summation, put into the rule. 
00-0 3-0 10-0 

3-2 18-0 25-2 

T2 = A III !^ 

^^ 67-6 = Q 
7T-4 = P 2 

i 135-2 = 2 Q 

309-6 = 4 P 

3-2 = A 
135-2 = 2Q 

448-0 = A + 4P + 2Q 



5-5 = 



2240 
2240 

3 )2464-0 

821-3 = A + 4 P + 2 Q x ~ = 

o 

sum of the moments of the half area of the load-water section 
reckoned from 1 ; the half area of the load-water section is 200*56 
feet (p. 478) ; the distance, therefore, of the centre of gravity of 
the load-water section from 1 will be found in intervals of space of 
5-5 feet, by dividing the sum of these moments by the half area, 
thus : — 

Half Area. Moments. No. 

200-56)821-3333 (4-09 intervals, each 
80224 5-5 ft. in length. 

190933 
180504 
10429 

and 4-09 X 5-5 = 22-5 ft. gives the distance of the centre of gravity 
of the load-water section from section 1 of the drawing. 

Relative capacities of the bodies immersed under the fore and 
after lengths of equal division of the load-water line-r— 
By former calculations. 

After-body immersed contains 497*79 cubic ft. of space. 

Fore-body " " 481-70 cubic ft. of space. 

Difference 16-09 = 

the excess in cubic feet of space of the body displaced under the 
after half-length of the load-water line over that under the fore- 
half of the same line — 

Sum of the bodies (by former calculation) or whole \ 070.49 

displacement in cubic feet of space (p. 477) j 

equal to 9-7949 hundreds of cubic feet of space, whence 16-09, or 
the difference between the two bodies in cubic feet, being divided 
by 9-7949, or the displacement expressed in terms of the hundreds 



482 



THE PRACTICAL MODEL CALCULATOR. 



of cubic feet of space, will give the excess for every hundred cubic 
feet of the whole displacement. 



Displacement in 
Hundreds of Cubic 



Excess in 
Cubic Feet 



Feet of Spa«B. of Space. 

9-7949 ) 16-09000(1-6 = Ratio of the excess of 
97949 the after-body of dis- 

629510 placement over the 

587694 fore-body of the same, 

•41816 denoted by a per-cent- 

age of the whole dis- 
METACENTRE. placement. 

A measure of the comparative stability of a ship, or the height 
of the metacentre above the centre of gravity of displacement esti- 



?/^dx 



in which /is the sign of in- 



mated, from the expression f / 
tegration and signifies sum : — 

y = the ordinates of the half-breadth load-water section. 

dx = the differential increment ofthe length of load-water section. 

D = displacement of the immersed portion of -the body in cu- 
bic feet of space. 



Ordinates from the table. 



Cubes of the Ordinates. 



•4 00-064 

3-0 27-000 

5-0 125-000 

6-0 216-00*0 

6^3 250-047 

6-1 226-981 

5-4 157-464 

3-7 50-653 

•4 0-064 

Cubes placed in O'Neill's rule for summation of 
Area = (A + 4P + 2Q) x ^ 



00-064 

00-064 

•128 


= A 


27-000 
216-000 
226-981 

50-653 
520-634 = P 


125-000 
250-047 
157-464 

532-511 = Q 

2 






4 

2082-536 = 4 P 

1065-022 = 2 Q 

000-128 = A 


1065-022 - 2 Q 






3147-686 = A + 4P + 2Q 
5-5 = r' 






15738430 
15738430 






St 


3)17312-2730 


-./ 




dx = 5770-7576 = A + 4P + 2Qx^ = 

o 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 483 

summation of the cubes of the ordinates of the load-water section ; 
and the height of the metacentre above the centre of gravity of 

displacement is expressed by f / , in which expression y'^ dx = 

5770*75 
5770-75 and D = 979-1 (p. 477) whence f x g^^.-^ = 3-98 feet 

is the height of the metacentre above the centre of gravity of the 
displacement. 

RESULTS OF THE CALCULATIONS. 

\8t Method. 
Displacement in cubic feet of space = 979-149. 
Displacement in tons of 35 cubic 1 _ 97-. 07-4 

feet of water to a ton j ~ 

Area of midship section.^ = 41-08 superficial feet. 

Area of load-water line or plane at "I _ . q-. .-. q ^ • i /• ^ 

the proposed deepest immersion.. ]~ ^ 

Tons to one inch of immersion at 

that flotation. 
Longitudinal distance of the centre^ 

of gravity of displacement from > = 22-22 feet. 

section 1. J 

Depth of the centre of gravity of^ 

displacement below the load-water > = 1-4904 feet. 

section j 

Distance of the centre of gravity of ^ 

the load-water section from verti- > = 22*5 feet. 

cal section 1 j 

Relative capacity of the after-body \ 

in excess of the fore-body in cubic V = 16*09 

feet of space j 



\ — -955 tons. 



-centage on the whole displace- \ _ -. ./ 
lent j 



ment 
Height of the metacentre above the^ 
centre of gravity of displacement, | 
estimated from the expression \ = 3-98 feet. 



f/V- J 



The young naval architect has thus been led through the essen- 
tial calculations on the immersed portion of a ship considered as a 
floating body. The term essential has here been used under a 
knowledge that the table of results might have been swollen to a 
small volume by a lengthened comparison of the elements of the 
naval construction, such as the ratio of the area of the midship sec- 
tion to the area of the load-water section, and that of the area of 
the midship section to the circumscribing parallelogram ; data that 
will always suggest themselves to the mind, and furnish salutary 
exercise for his judgment, while the introduction of such com- 
parisons into these rudiments might deter the novice from entering 



484 



THE PRACTICAL MODEL CALCULATOR. 









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§ 


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§ s 








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10-20 


19^15 


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28-15 


Longitudinal 
Areas 106-50. 




Multipliers _^ 
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1 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 



485 





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andAfteiv j^],. 
bodies. II pl'^.V 




1 








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A 


6-3 


3-15 


6-1 


305 


5-4 


I 
2-70 1 3 7 


1-85 


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14-171 97.0,' 1 "'°*' 
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Functions 

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110-75 217-25 .3.53-72 


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486 THE PRACTICAL MODEL CALCULATOR. 

on a task that would thence seem to be involved in such voluminous 
results. For the second method of calculation, the table of ordi- 
nates is in two portions, viz. the fore and after-bodies under the 
division of the load-water section into two equal parts, the length 
of such section being restricted to the distance from the fore-edge 
of the rabbet of the stem to the after-edge of the rabbet of the 
post. The enlarged tables are shown at pages 484 and 485, and 
the directions for the working of these tables have been given at 
page 459, observing only that the ordinates have not been herein 
inserted in red, as it was there suggested, to insure perspicuity and 
accuracy. 

RESULTS FROM THE TABLES. 

{A ^ 2r 

2 + 2P + Q I y 

And solid = areas for ordinates 1 _ / ^' , 9 p/ i n/ \ i ^ 
summed by rule f -\ 2 '^ ^^ '^ "^ j '^ 3 

Functions of the areas marked B=< 2" + 2I*~l~Qf 

Function of the solid equal to B, placed in O'Neill's rules = 
A' + 2 P' + Q' = E 

2r 2r' 
Whence displacement = E x -h- x -o-, in the example r = -92 

r' = 5-5. 

2r 2r' ^ 1-84 11 
Therefore J displacement = E x -k- x —^ = E x — g- x ^ = 

20-24 

VALUE OF E FROM THE TABLES BY VERTICAL SECTIONS. 

Table 1... 106-50 = submultiple of the fore-body by vertical sections. 

Table 2. .. 110-77 = " after-body 

217-27 = sum of the submultiples = E. 

_ 20-24 217-27 x 20-24 ^, , , ^, ^, 
J displacement = E x — ^ = g = 24-14 X 20-24 = 

488-5936 = J solid of displacement by the summation of the 
2 vertical areas given in cubic feet of space. 

5) 977-1872 
7) 195-4374 

27*92 = Displacement by vertical sections in tons of 35 
cubic feet of space. 

VALUE OF E FROM THE TABLES BY HORIZONTAL SECTIONS. 

Table 1... 106 -50 = submultiple of the fore-body by horizontal 

• sections. 
Table 2... 110 -75 = submultiple of the after-body by horizontal 

sections. 

From whence the same results will be obtained. 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 487 



AREA or MIDSHIP SECTION. 

From table 1...28'15 = submultiple of the area of Section 5. 
1-84 = 2r 



11260 
22520 
2015 

3)51-7960 

17*265 = J area of upper space of midship section. 
3-276 = J area of the lower " " below d d, 

20-540 = J area of midship section. 

2 



41-08 = area of midship section. 

AREA OF THE LOAD-WATER LINE. 

From table 1.. .26-35 = submultiple of the area of the fore-body. 
From table 2. ..28-35 = " " after-body. 

54-70 = submultiple for J area of load-water line. 
11 = 2 / 



3)601-7 

A 2r' 

200-56 = J area ==2+^^ + Q^~3~ 



12 ) 401-12 = area of load-water line. 
5)33-42 



7)6-684 

•955 = tons per inch of immersion at the load- 
water line. 

POSITION OF THE CENTRE OF GRAVITY OF DISPLACEMENT. 

By table 2. ..878-86 = moments from Section 1. 

and E 217-27 = corresponding function of the displacement. 

217-27) 878-86 (-404 intervals of ^-^ feet, giving 4-04 x 
869-08 5-5 = 22-22 feet as the distance 

of the centre of gravity of the dis- 

97800 placement from Section 1. 

86908 



10892 



488 THE PRACTICAL MODEL CALCULATOR. 

DEPTH OF THE CENTRE OF GRAVITY OF THE DISPLACEMENT BELOW THE 
LOAD-WATER SECTION. 

By table 2. ..353*72 = moments from load-water line. 

and E 217'25 = corresponding function of the displacement. 

217-25) 353-72 (1-62 intervals of -92 feet, giving 1-62 x 
217-25 -92 = 1-4904 as the distance that 

136-470 *^® centre of gravity of displace- 

130-350 ment is below the load-water line. 

61200 
43450 

17750 • 

POSITION OF THE CENTRE OF GRAVITY OF THE LOAD-WATER LINE OP 
DEEPEST IMMERSION. 

From table 1 26-35 ft. From table 2. ..224-000 = moments 

2 28-35 from 1st section. 

Function for area..54-7) 224-0 (4-09 intervals of 5-5 feet, giving 

218-8 4-09 X 5-5 = 22-495 feet 

5200 ^^ ^^^ distance that the 

4923 centre of gravity of the 

load-water section is from 

•277 vertical section 1. 

RELATIVE CAPACITIES OF THE CALCULATED IMMERSED BODIES CON- 
TAINED UNDER THE FORE AND AFTER-LENGTHS OF EQUAL DIVISION 
OF THE LOAD-WATER LINE. 

Feet. 

From table 1... Function for the fore-solid 106-50 

From table 2... Function for the after-solid 110-75 

4-25 

Sum of the functions 217-25 

The difference, 4*25 feet, expresses the excess in cubic feet of space 
of the body, displaced under the after half-length of the load-water 
line, over that under the fore half-length of the same line, and the 
sum of the functions, 217*25, is equal to 2-1725 hundreds of cubic 
feet of space ; whence, 4-25 feet, or the difference between the 
functions for the two bodies, being divided by the function 2-1725, 
or the function for the displacement of the calculated body ex- 
pressed in terms of hundreds of cubic feet of space, will give the 
excess for every hundred cubic feet of that displacement : 

Function of Excess in 

Displace- Cubic Feet 

meat. of Space. 

2-1725)4-25000(1-9 ratio of the excess of the after- 
2-1725 body of calculation over the 

207750 fore-body of the same, de- 

1955'^5 noted by a per-centage of the 

^ displacement calculated by 

•12225 the table of ordinates. 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 489 

HEIGHT OF THE METACENTRE ABOVE THE CENTRE OF GRAVITY OF 
DISPLACEMENT. 

From table 2... The summation of the functions^ 

of the cubes of the ordinates for the value of > = 1573*843. 
the / y^ dx j 

The corresponding function for the solid = 217*25. 

from whence the height of the metacentre above the centre of 

2 y^ dx 
gravity of displacement, expressed by q- / is as follows : 

fy^dix = 1573*843 x -g- where r' = 5*5 feet = 

1573*843 X 11 17312*273 ^^^^ ^^ ^ 
3 = 3 = 5770*75 feet. 



2r 2 



(Page 485) 217*27 x -g- x -g- = J displacement = 488*5936 feet, 
whence displacement or D = 977*1872 ; and thence 
2 ^^_2 5770-75 11541*53 
3*^ D ~ 3 ^ 977-1872 ~ 2931*5616 ~ ^'^^ ^^^*' 

RESULTS OBTAINED UNDER THE TWO METHODS OF CALCULATION 
CONTRASTED. 

Old Method. Second Method. 

Displacement in cubic feet of space... 979*139 977*187 

Displacement in tons of 35 cubic feet 

of water to a ton 27*985 27*92 

Superficial ft. Superficial ft. 

Area of midship section 41*08 41*08 

Area of load-water line or plane at 

the proposed deepest immersion 401*12 401*12 

Tons to one inch of immersion at line 

of flotation '9526 tons. '955 tons. 

Longitudinal distance of the centre of 
gravity of the displacement from 
section 1 22*22 ft. 22*22 ft. 

Depth of the centre of gravity of dis- 
placement below the load-water sec- 
tion 1*4812 ft. l*4904ft. 

Relative capacities of the bodies 1*6 per cent. l*9perct. 

Height of the metacentre above the 

centre of gravity of displacement... 3*98 ft. 3*98 ft. 

THIRD METHOD OF CALCULATION. 

CALCULATIONS ON THE DRAUGHT OF THE YACHT OF 86 TONS USING THE 
CURVE OF SECTIONAL AREAS. 

The load-water line AB, in the sheer plan, is divided into 
two equal parts at the point C, and those equal parts are again 
subdivided at the points D and E ; at the points C, D, and E, 



490 



THE PRACTICAL MODEL CALCULATOR. 




St-C 



!•* 



tS 



RH = 2-4 feet. 
QI = 4-1 " 
PK = 2-45 " 



Ordinates. 



DN = 5-8 feet. 
CM = 5-0 " 
EO = 4-2 " 



AB = 44 feet. 
FG = 44 " 
FI = 22 « 



IG = 22 feet. 
QG = 22-87 " 
FQ = 22-37 " 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 491 

thus obtained, the transverse vertical sections of the vessel are 
delineated. 

The length of the load-water line from the fore edge of the rab- 
bet of the stem B, to the after edge of the rabbet of the post A, is 
next drawn below and parallel to the base line SF of the sheer plan ; 
this line, FG, becomes the base line of the curve of the sectional 
areas. The common sections of the transverse vertical sections 
of C, D, and E, (which will be straight lines,) with this horizontal 
and longitudinal plan, are drawn from their respective points of 
division, H, I, and K, in half-breadth plan. The areas of these 
transverse vertical sections at D, C, and E, are then calculated, 
as before, thus : — 

Area =|A + 4P-f2Q|xg=|2+2P-f Q|x-^;or, 

{"i 3 r A 1 3 

A -f 2P + 3Q lxgr=^ -^-\-V + 1'5QIx ^7\ 

Half Area of Transverse Vertical Section, at 0, by Rule 1, 

{A 1 2 r 

2" + 2P 4- Q Vx -g-. 

1st. ...6-3 2d ...6-0 3d.. .4-8 = Q 

Last... -2 4th. ..2-3 



2)6-5 8-3 = P 

A 
2 



3-25 =4 2 



16-60 = 2P 
3-25 = ^ 

4-80 = Q 



24-65= 2 + 2P -f Q 

on 2r 
•83 =-^ 



7395 
19720 

20-4595 =^-f2P-fQx-^=J area 

of section C in feet. 

CM 5-0 

= 1-25 = r = 



CM, or depth = 5-0 feet, whence -j-, or -j 



,2r 2x1-25 2-5 ^^^ 
distance between the ordinates, and -q- = g = -g- = -oo teet. 



II?} 4+^+1-^^ 



492 THE PRACTICAL MODEL CALCULATOR. 

Ealf Area of Section Q, hy Rule 2, 
or, -J area =1 2 + P + 1-5 Q I X ^ r. 
1st 6-3 P = ^'Q 2d. 

La;t:•;^2 ^ 3d. 

ir^-^ _^2 = iQ. 

^ "^^ "" 2 12-97 = 1-5 Q. 

3-25 
12- 

16-22 

5 = 3 r = CM = 5-0 feet. 

4) 8140 j—^ g 

20-275 =Jarea= 2 +P + 1-5Q x^^. 
Ealf Area of the Transverse Vertical Section at U. 
l3t. ...5-0 2d. ...4-2 8d. ...2-9 = Q 

Last... -2 4th. ...Vl 

2)^ 5-9 = P 

^^ 2 lT8 = 2P 

A 
2-6=2 
2-9 = Q 

rr3=|^ + 2P + Q 

EO 4-2 ^ ^^ ,. 

EO, or depth = 4-2 feet, whence -4- = ^ = 1'05 = ?* = dis- 

2r 1-05x2 2-1 _ 
tance between the ordmates, and -g- = 3 — 3 " ' ^^^^ > 

tVipreiore 

Area 1{| + 2P + q}x ^ = 17-3 x -7 = 12-11 = half 

area of transverse vertical section at E. 

Ealf Area of the Transverse Vertical Section at D. 
1st. ...5-40 2d. ...3-5 3d. ...1-46 = Q 

Last. ..9-2 4th.. ..0^7 

2)576" ^ 4-2 = P 

2'« - 2 84 = 2P 

A 
2-8=2 

1-46 = Q 
12-66 = 1^ + 2P + Q 



SHIP-BUILDING AND NAVAL ARCHITECTURE. 493 

DN 5-8 
DN, or depth = 5-8 feet, whence —j- = -j- = 1*45 feet = 

, ,. 27^ 2 X 1-45 2-9 
r = distance between the 6rdinates, and -q- = q = ~o" == 

•97 feet ; therefore, 

Area =|2-2 + P+q|x^ = 12-66 X -97 = 12-28 feet = 

half area of transverse vertical section at T>. 

Half Areas of the Transverse Vertical Sections. 

Feet. Feet. 

rE = 12-11^ Divided by 5 as the depth assumed for r2-42 
At< C = 20-20 y the zone, give the ordinates for the curves 4*04 

(D = 12-28 J of sectional areas, as (2-45 

of which 2-42 is set off from H as HR, 4-04 feet from I as IQ, 
and 2-45 feet from K as KP ; the curve IRQPG, passing through 
the extremities P, Q, and R of the ordinates PK, QI, and RH, is 
the curve bounding the area of a zone, which, to the depth of 5 feet 
for a solid, will give in cubic feet of space the half displacement 
of the immersed body, or the displacement of the yacht to the line 
AB of proposed deepest immersion. * 

To measure this representative area, and from thence the solid, 
join the points Q, Gr, and I by the straight lines QG, QF, dividing 
the curvilinear area FRQPGF into the two triangles QGI, QFI, 
and the two areas GPQG, FRQF. The triangles by construction 
are equal, and the area of each one of them is equivalent to 
GI X QI GI X QI 

2 » 0^ ^^® ^^^^^ a^®^ GQFIG = 2"^ X 2 = GI X QI 

or FI X IQ, FI being equal to IG, each being the half-length of 
the same element, the load-water line or line of deepest immersion. 
The areas QPGQ, QRFQ, are bounded by the curve lines QPG, 
QRF, which are assumed as portions of common parabolas, and 
under such an assumption their respective areas are equal to f of 
the circumscribing parallelograms, or the area QPGQ = f of 
GQ X X, and the area FRQF = f of FQ X x^, where x and x' 
are the greatest perpendiculars that can be drawn from the bases 
QG and QF to meet the curves QPG, QRF. 

DISPLACEMENT. 

AB by a scale of parts = 44 feet, whence FI or IG equal 

AB 44 

-5- = -^ feet = 22 feet; ordinate QI of the medial section = 

4-04 feet ; and QG = FQ, being the respective hypothenuses of the 
equal triangles QGI, QFI, are each equal to -v/IG^ + QP = 

v/222 -r 4^ = ^/484 + 16-32 == n/500-32 = 22-37 feet ; and 
X, by measurement with a scale of parts, = -6 foot, and x^ also 
•6 foot, from which the half displacement in cubic feet of space will 
be obtained as follows : — . 



494 THE PRACTICAL MODEL CALCULATOR. 

Area FQGIF = GI x IQ. cuwc feet. 

Solid under the \ = (ji x IQ x 5 = 22 X 4-1 X 5 = 451-00 

area i^ Qliir j 
Area QPGIQ = f of GQ X a; 

Solidunderthe|^2of (^Qx a; X 5 = 1x22-37 X -6x5= 44-74 
area QPGQ J a 

Area FRQF = | of FQ x a;' 

Solid under the l,^fj,Q^^,^5_^^_22-37x-6x5= 44-74 
area r KQi! i a -^ 6 

540-48 

or area of the triangle QGI + area of the triangle QFI + area 
of the space QPGQ + area of the space FRQF = to the repre- 
sentative area FRQPG, which being multiplied by the assumed 
depth of 5 feet for the zone of half displacement gives 540-48 cubic 
feet of space, which divided by 35, as the number of such cubic 
feet that are equivalent to one ton of medium water, gives 

3)540-48 

7)108-09 



15-44 tons for half displacement, 

and that the whole weight of the body is equal to 15-54 X 2 = 
30-88 tons = displacement to the line of proposed deepest immer- 



sion 



AB. 



RELATIVE CAPACITIES OF THE BODIES IMMERSED UNDER THE FORE AND 
AFTER HALF-LENGTHS OF THE LOAD-WATER LINE, AS GIVEN BY THE 
DELINEATED CURVE OF SECTIONAL AREAS. 

The triangles QGI and QFI being equal, the relative capacities 
of the fore and after-bodies will be determined by the proportion 
that the area QPGI bears to the area QRFI ; and as these areas 
involve two equal terms, or that the base FQ = the base QG, it 
follows, that the relation of these areas to each other will be ex- 
pressed by the proportion that the perpendiculars x and x' bear 
to each other. In the example given, the fore and after-bodies, or 
the displacements under the fore and after half-lengths of the load- 
water AB, are equal ; as the perpendiculars x and x' taken from 
the diagram, on a scale of equal parts, are each -6 of a foot. 

The area of the midship section is denoted relatively by the 
medial ordinate of the curve of sections QI, and the full amount 
of it is obtained by multiplying the function QI by the depth of 
the zone M. In the example : 

M = 5 ; QI = 4-04 ; then half area of medial section = 4-04 X 5 

5 



Area of midship section 20-20 



TABLES OF LOGARITHMS. 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Kii 


1 

No. 


1 

Los. 


P.OP. 

Part. 


No. 


Log. i 


Prop.' 
Part. 


No. 


Log. 


Prop, j 
Part. 




1000 


000000 


11 


1000 


025306 




1120 


049218 


1 


1180 


071882 




1 


000484 


43 i 


1 025715 


41 


1 


049606 


39 1 


1 ! 072250 


37 




2 


000868 


86 1 


2 ! 026124 


82 


2 


049993 


7^ 1 


2 U72617 


73 




3 


001301 


130 j 


3 026533 


122 


3 


050380 


1161 


3 i 072985 


110 




4 


001734 


173 


4 1 026942 


163 


4 


050766 


154 


4 1 073352 


147 




5 


002166 


216! 


5 027350 


204 


5 


051152 


193! 


5 


073718 


183 




6 


002698 


259 


6 027757 


245 


6 


051538 


232 


6 


074085 


220 




7 


003029 


303 


7 ■ 028164 


286 


7 


051924 


270 


7 


074451 


256 




8 


003460 


346 1 


8 1 028571 


326 


8 


052309 


309 


8 


074816 


293 




9 


003891 


389 !i 


9 028978 


367 


9 


052694 


347 


9 


075182 


330 




1010 


004321 


i 


1070 


029384 




1130 


053078 




1190 076547 






1 


004751 


43 i 


1 


029789 


40 


1 


053463 


88 


1 1 675912 


36 




2 


005180 


86! 


2 


030195 


81 


2 


053846 


77 


2 1 076276 


73 




3 


0U56U'.J 


128 1 


3 


030600 


121 


3 


054230 


115 


3 


076640 


109 




4 


006038 


171 


4 


031004 


162 


4 


054613 


153 


4 


077004 


145 




6 


006466 


214 1 


5 


031408 


202 


5 


054996 


191 ! 


5 


077368 


181 




6 


006894 


257 ; 


6 


031812 


242 


6 


055378 


230 i 


6 


077731 


218 




7 


00732] 


300 


7 


032216 


283 


7 


055760 


268; 


7 


078094 


254 




8 


007748 


343 


8 


032619 


323 


8 


056142 


306 


8 


078457 


290 




9! 


008174 


385, 


9 


033021 


364 


9 


056524 


3451 


9 


078819 


327 




1020 


008600 


1 


1080 


033424 




1140 


056905 


1 


1200 


079181 






1 


009026 


42! 


1 


033826 


40 


1 057286 


38 1 


1 


079543 


36 




2 


000451 


85 1 


•) 


034227 


80 


2 


057666 


76 1 


2 I 079904 


72 




3 


009876 


127 i 


3 


034628 120! 


3 


058046 


114 


3 


080266 


108 




4 


010300 


170 1 


4 


035029 1601 


4 


058426 


152 


4 


080626 


144 




5 


010724 


212 i 


5 


035430 


200 


5 


058805 


190 


6 


080987 


180 




6 


011147 


254 i 


6 


035830 


240 


6 


059185 


228 


6 1 081347 


216 




7 


011570 


297 


7 


036229 


280 


7 


059563 


266 


7 i 081707 


252 




8 


011993 


339 


8 


036629 


321 


8 


059942 


304 


8 j 082067 


288 




9 


012415 


382 


9 


037028 


361 


9 


060320 


342 


9 ! 082426 


324 




1030 


012837 




1090 


037426 




1150 


060698 




1210 


082785 






1 


013259 


42 


1 


037825 


40 


1 


061075 


38 


1 


083144 


36 




2 


013680 


84 


2 


038223 


79 


2 


061452 


75 


2 


083503 


71 




3 


014100 


126 


3 


038620 


119 


3 


061829 


113 


3 


083861 


107 




4 


014520 


168 


4 


039017 


159 


4 


062206 


160 


4; 084219 


143 




5 


014940 


210 


5 


039414 


198 


5 


062582 


188 


5 


084576 


179 




6 


015360 


252 


6 


039811 


238 


6 


062958 


226 


6 


084934 


214 




7 


015779 


294 


7 


040207 


278 


7 


I 063333 


263 


7 


085291 


250 




8 


016197 


336 


8 


040602 


318 


8 


063709 


301 


8 


085647 


286 




9 


016615 


378 


9 


040998 


357 


9 


064083 


338 


9 


086004 


322 




1040 


017033 




1100 


041393 


i 


1160 


064458 




il220 


086360 






1 


1017451 


42 


1 


041787 


39 


1 


064832 


37 


1 


086716 


35 




2 


017868 


83 


2 


042182 


79 


2 


065206 


75 


2 i 087071 


71 




3 


018284 


425 


3 


042575 


118 


3 


065580 


112 


i 3 1 087426 


106 




4 


018700 


166 


4 


042969 


157 


4 


: 065953 


149 


4 1 087781 


142. 




5 


019116 


208 


5 


043362 


196 


5 1 066326 


186 


] 5 \ 088136 


177 




6 


019532 


250 


6 


043755 


236 


6 1 066699 


224 


j 6 i 088490 


213 




7 


019947 


291 


7 


044148 


275 


7 j 067071 


261 


1 7 : 088845 


248 




8 


020361 


333 


8 


044540 


314 


8 1 067443 


298 


1 8 i 089198 


284 




9 


020775 


374 


9 


044931 


354 


9 : 067814 


336 


9 1 089552 


319 




1050 


021189 


! 


1110 


045323 




1170 068186 


1 


1230 089905 






1 


021603 


41 


1 


045714 


i 39 


1 ! 068557 


1 37 


1 090258 


35 




2 


022016 


82 


2 


046105 


1 78 


2 1 068928 


1 74 


2 090611 


70 




3 


022428 


124 


3 


046495 


117 


3 1 069298 


1111 


3 1 090963 


106 




4 


022841 


165 


' 4 


046885 


1 156 


4 069668 


1148 


41091315 


141 




5 


023252 


206 


5 


047275 


i 195 


5 I 070038 


|185 


5 091667 


176 




6 


023664 


247 


6 


! 047664 


I234 


6 ! 070407 i 222 


6 1 092018 


211 




7 


024075 


288 


i 7 


i 018053 


273 


7 ! 070776 I 259 


7 : 092370 


246 




8 


. 024486 


330 


I 8 


i 048142 


312 


8 j 071145 1 296 


1 8 


092721 


282 




9 


: 024896 


371 


1 9 048830 


1 351 


9 071514 333 


\. 9 


093071 


317 



32 



LOGARITHMS OF NUMBERS. 



No. 


i^og. i^-?; 


No. i Log. 


Mi 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


1 

1 1240 


098422 




1300 118948 




1360 


183539 




1420 


152288 




1 


093772 


35 


1 


114277 


88 


1 


138858 


32 


1 


152594 


30 


2 


094122 


70 


2 


114611 


67 


2 


184177 


64 


2 


152900 


61 


3 


094471 


105 


3 


114944 


100 


8 


134496 


96 


3 


153205 


91 


4 


094820 


140 


4 


115278 


133 1 


4 


134814 


127-1 


4 


153510 


122 


5 


095169 


175 


6 


115610 


1671 


6 


135133 


159! 


5 


153815 


152 


6 


095518 


210 


6 


115943 


200 


6 


185451 


191 j 


6 


154119 


183 


7 


095866 


245 


7 


116276 


233 1 


7 


135768 


223 i 


7 


154424 


213 


8 


096215 


280 


8 


116608 


267, 


8 


136086 


255 


8 


154728 


244 


9 


096562 


315 


9 


116940 


800 


9 


136403 


287 


9 


155032 


274 


1250 


096910 




1310 


117271 




1870 


186721 




1430 


155836 




1 


097257 


85 


1 


117603 


33' 


1 


137037 


32 


1 


155640 


30 


2 


097604 


69 


2 


117984 


66! 


2 


137354 


63 


2 


155948 


60 


3 


097951 


104 


3 


118265 


99' 


3 


137670 


94 


3 


156246 


91 


4 


098297 


138 


4 


118595 


132 


4 


137987 


126 


4 


156549 1 121 


5 


098644 


173 


5 


118926 


165 


5 


138303 


158 


5 


156852 1 151 


6 


098990 


208 


6 


119256 


198 j 


6 


138618 


189 


6 


157154 


181 


7 


099335 


242 


7 


119586 


231, 


7 


188934 


221 


7 


157457 


211 


8 


099681 


277 


8 


119915 


264' 


8 


139249 


252 


8 


157759 


242 


9 


100026 


311 


9 


120245 


297 


9 


139564 


284 


9 


158061 


272 


1260 


100370 




1320 


120574 




1380 


139879 




1440 


158862 




1 


100715 


34 


1 


120908 


33 


1 


140194 


31 


1 


158664 


30 


2 


101059 


69 


2 


121231 


66 


2 


140508 


63 


2 


158965 


60 


3 


101403 


103 


3 


121560 


98 


3 


140822 


94 


3 


159266 


90 


4 


101747 


187 


4 


121888 


131 


4 


141136 


125 


4 


159567 


120 


5 


102090 


172 


5 


122216 


164 


. 6 


141450 


157 


5 


159868 


150 


6 


102434 


206 


6 


122543 


197 


6 


141763 


188 


6 


160168 


180 


7 


102777 


240 


7 


122871 


230 


7 


142076 


219 


7 


160468 


210 


8 


103119 


275 


8 


123198 


262 


8 


142389 


251 


8 


160769 


240 


9 


103462 


309 


9 


123525 


295 


; 9 


142702 


282 


9 


161068 


270 


1270 


103804 




1330 


128852 




1390 


143015 




1450 


161368 




1 


104146 


34 


1 


124178 


88 


1 


143327 


31 


1 


161667 i 30 I 


2 


104487 


68 


2 


124504 


65 


2 


143639 


62 


2 


161967 


60 


3 


104828 


102 


3 


124830 


98 


3 


148951 


93 


3 


162266 


89 


4 


105169 


136 


4 


125156 


130 


4 


144263 


125 


4 


162564 


119 


5 


105510 


170 


6 


125481 


163 


5 


144574 


156 


5 


162863 


149 


6 


105851 


204 


6 


125806 


195 


6 


144885 


187 


6 


168161 


179 


7 


106191 


238 


7 


126131 


228 


7 


145196 


218 


7 


163460 


209 


8 


106581 


272 


8 


126456 


260 


8 


145507 ! 249 


8 


163757 


239 


9 


106870 


306 


9 


126781 


293 


9 


145818 


280 


9 


164055 


269 


1280 


107210 




1340 


127105 




1400 


146128 




1460 


164353 




1 


107549 


34 


1 


127429 


32 


1 


146488 


31 


1 


164650 


30 


2 


107888 


67 


2 


127752 


65 


2 


146748 


62 


2 


164947 


59 


3 


108227 


101 


3 


128076 


97 


8 


147058 


93 


i3 


165244 


89 


4 


108565 


185 


4 


128399 


129 


4 


147367 


124 


4 


165541 


119 


5 


108903 


169 


6 


128722 


161 


5 


147676 1 155 


6 


165838 


148 


6 


109241 


203 


6 


129045 


194 


6 


147985 ! 186 


6 


166134 


178 


7 


109578 


237 


7 


129368 


226 


7 


148294 


217 


7 


166430 


207 


8 


109916 


270 


8 


129690 


258 


8 


148608 


248 


8 


166726 


237 


9 


110258 


304 


9 


130012 


291 


9 


148911 


279 


9 


167022 


267 


1290 


110590 




1350 


180334 




1410 


149219 




1470 


167317 




1 


110926 


34 


1 


130655 


32 


1 


149527 


31 


I 


167613 


29 


2 


111262 


67 


2 


130977 


64 


2 


149835 


61 


2 


167908 


59 


8 


111598 


101 


8 


131298 


96 


3 


150142 


92; 


3 


168203 


88 


4 


111934 


134 


4 


131619 i 128 


4 


150449 


128 1 


4 


168497 


118 


5 


112270 


168 


! 5 


131989 ! 160 


5 


150756 


154: 


6 


168792 


147 


6 


112605 ! 201 


6^ 


182260 192 


6 


151063 


184 


6 


169086 


177 


7 


1129401 235 


7 


132580 224 


7 


151370 


215 


7 


169380 


200 j 


8 


118275 268 


8 


182900 256 


8 


151676 


246 


8 


169674 


236 


9 


118609 1 302 


9 


138219 288 


9 


151982 


277 


9 


169968 I 265 | 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


1480 


170262 




1540 


187521 




1600 


204120 




1660 


220108 




1 


170555 


29 


1 


187803 


28 


1 


204391 


27 


1 


220370 


26 


2 


170848 


58 


2 


188084 


56 


2 


204662 


54 


2 


220631 


52 


3 


171141 


88 


3 


188366 


84 


3 


204933 


81 


3 


220892 


78 


4 


171434 


117 


4 


188647 


113 


4 


205204 


108 


4 


221153 


104 


5 


171726 


146 


6 


188928 


141 


5 


205475 


135 


5 


221414 


130 


6 


172019 


175 


6 


189209 


169 


6 


205745 


162 


6 


221675 


157 


7 


172311 


204 


7 


189490 


197 


7 


206016 


189 


7 


221936 


183 


8 


172603 


234 


8 


189771 


225 


8 


206286 


216 


8 


222196 


209 


9 


172895 


263 


9 


190051 


253 


9 


206556 


243 


9 


222456 


235 


1490 


173186 




1550 


190332 




1610 


206826 




1670 


222716 




1 


173478 


29 


1 


190612 


28 


1 


207095 


27 


1 


222976 


26 


2 


173769 


58 


2 


190892 


56 


2 


207365 


54 


2 


223236 


52 


3 


174060 


87 


3 


191171 


84 


3 


207634 


81 


3 


223496 


78 


4 


174351 


116 


4 


191451 


112 


4 


207903 


108 


4 


223755 


104 


5 


174641 


145 


5 


191730 


140 


5 


208172 


135 


5 


224015 


130 


6 


174932 


175 


6 


192010 


168 


6 


208441 


162 


6 


224274 


156 


7 


175222 


204 


7 


192289 


196 


7 


208710 


188 


7 


224533 


182 


8 


175512 


233 


8 


192567 


224 


8 


208978 


215 


8 


224792 


208 


9 


175802 


261 


9 


192846 


252 


9 


209247 


241 


9 


225051 


284 


1600 


176091 




1560 


193125 




1620 


209515 




1680 


225309 




1 


176381 


29 


1 


193403 


28 


1 


209783 


27 


1 


225568 


26 


2 


176670 


58 


2 


193681 


56 


2 


210051 


54 


2 


225826 


52 


3 


176959 


86 


3 


193959 


83 


3 


210318 


80 


3 


226084 


77 


4 


177248 


115 


4 


194237 


111 


4 


210586 


107 


4 


226342 


103 


5 


177536 


144 


5 


194514 


139 


5 


210853 


134 


5 


226600 


129 


6 


177825 


173 


6 


194792 


166 


6 


211120 


161 


6 


226858 


155 


7 


178113 


202 


7 


195069 


194 


7 


211388 


187 


7 


227115 


181 


8 


178401 


231 


8 


195346 


222 


8 


211654 


214 


8 


227372 


206 


9 


178689 


259 


9 


195623 


250 


9 


211921 


240 


9 


227630 


232 


1510 


178977 




1570 


195900 




1630 


212188 




1690 


227887 




1 


179264 


29 


1 


196176 


27 


1 


212454 


27 


1 


228144 


26 


2 


179552 


57 


2 


196452 


55 


2 


212720 


53 


2 


228400 


51 


3 


179839 


86 


3 


196729 


83 


3 


212986 


80 


3 


228657 


77 


4 


180126 


115 


4 


197005 


110 


4 


213252 


106 


4 


228913 


102 


5 


180413- 


144 


5 


197281 


138 


5 


213518 


133 


5 


229170 


128 


6 


180699 


172 


6 


197556 


166 


6 


213783 


159 


6 


229426 


154 


7 


180986 


201 


7 


197832 


193 


7 


214049 


186 


7 


229682 


179 


8 


181272 


230 


8 


198107 


221 


8 


214314 


212 


8 


229938 


205 


9 


181558 


258 


9 


198382 


248 


9 


214579 


239 


9 


230193 


231 


1520 


181844 




1580 


198657 




1640 


214844 




1700 


230449 




1 


182129 


28 


1 


198932 


27 


1 


215109 


26 


1 


230704 


25 


2 


182415 


57 


2 


199206 


55 


2 


215373 


53 


2 


230960 


51 


3 


182700 


86 


3 


199481 


82 


3 


215638 


79 


3 


231215 


76 


4 


182985 


114 


4 


199755 


110 


4 


215902 


106 


4 


231470 


102 


5 


183270 


143 


5 


200029 


137 


5 


216166 


132 


5 


231724 


127 


6 


183554 


171 


6 


200303 


164 1 


6 


216430 


158 


6 


231979 


153 


7 


183839 


200 


7 


200577 


192 


7 


216694 


185 


7 


232233 


178 


8 


184123 


228 


8 


200850 


219 


8 


216957 


211 


8 


232488 


204 


9 


184407 


256 


9 


201124 


247 


9 


217221 


238 


9 


232742 


229 


1530 


184691 




1590 


201397 




1650 


217484 




1710 


232996 




1 


184975 


28 


1 


201670 


27 


1 


217747 


26 


1 


233250 


25 


2 


185259 


57 


2 


201943 


54 


2 


218010 


52 


2 


233504 


51 


3 


185542 


85 


3 


202216 


82 


3 


218273 


79 


3 


233757 


76 


4 


185825 


113 


4 


202488 


109 


4 


^18535 


105 


4 


234011 


101 


5 


186108 


142 


5 


202761 


136 


5 


218798 


131 


5 


234264 


127 


6 


186391 


170 


6 


203033 


163 


6 


219060 


157 


6 


234517 


152 


7 


18G674 


198 


7 


203305 


191 


7 


219322 


183 


7 


234770 


177 


8 


186956 


227 


8 


203577 


218 


8 


219584 


210 


8 


235023 


202 


9 


187239 


255 


9 


203848 


245 


9 


219846 


236' 


9 


235276 


228 



LOaARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Pan. 


No. 


Log. 


Prop. 

Part. 


No. 


Log. 


Prop. 
Pare. 


No. 


Log. 


Prop, i 

Part, j 


1720 


235528 




1780 


250420 




1840 


264818 




1900 


278754 




1 


285781 


25 


1 


250664 


24 


1 


265054 


23 


1 


278982 


23 


2 


236033 


50 


2 


250908 


49 


2 


265290 


47 


2 


279210 


45 


3 


236285 


76 


3 


251151 


73 


3 


265525 


70 


3 


279439 


68 


4 


236537 


101 


4 


251395 


97 


4 


265761 


94 


4 


279667 


91 


5 


236789 


126 


5 


251638 


121 


5 


265996 


117 


5 


279895 


114 


6 


237041 


151 


6 


251881 


146 


6 


266232 


141 


6 


280123 


137 


7 


237292 


176 


7 


252125 


171 


7 


266467 


164 


7 


280351 


160 


8 


237544 


2U2 


8 


252367 


195 


8 


266702 


188 


8 


280578 


182 


9 


237795 


227 


9 


252610 


219 


9 


266937 


211 


9 


280806 


205 


1730 


238046 




1790 


252853 




1850 


267172 




1910 


281033 




1 


238297 


25 


1 


253096 


24 


1 


267406 


23 


1 


281261 


23 


2 


238548 


50 


2 


253338 


48 


- 2 


267641 


47 


2 


281488 


45 


3 


238799 


75 


3 


253580 


73 


3 


267875 


70 


3 


281715 


68 


4 


239049 


100 


4 


253822 


97 


4 


268110 


94 


4 


281942 


91 


5 


239299 


125 


5 


254064 


121 


5 


268344 


117 


5 


282169 


113 1 


6 


239550 


150 


6 


254306 


145 


6 


268578 


141 


6 


282395 


136 


7 


239800 


175 


7 


254548 


170 


7 


268812 


164 


7 


282622 


159 i 


8 


240050 


200 


8 


254790 


194 


8 


269046 


188' 


3 


282849 


181 


9 


240300 


225 


9 


255031 ! 218 


9 


269279 


211 


9 


283075 


204 


1740 


240549 




1800 


255273 




1860 


269513 




1920 


283301 




1 


240799 


25 


1 


255514 


24 


1 


269746 


. 23 


1 


283527 


23 


2 


241048 


50 


2 


255755 


48 


2 


269980 


47 


2 


283753 


45! 


3 


241297 


75 


3 


255996 


72 


3 


270213 


70 


3 


283979 


68 


4 


241546 


100 


4 


256236 


96 


4 


270446 


93 


4 


284205 


90 


5 


241795 


-124 


5 


256477 


120 


5 


270679 


116 


5 


284431 


113 


6 


242044 


149 


6 


256718 


144 


6 


270912 


140 


6 


284656 


135 


7 


242293 


174 


7 


256958 


168 


7 


271144 


163 


7 


284882 


158 


8 


242541 


199 


8 


257198 


192 


8 


271377 


186 


8 


285107 


180 


9 


242790 


223 


9 


257439 


216 


9 


271609 


210 


9 


285332 


203 


1750 


243038 




1810 


257679 ■ 


1870 


271842 




1930 


285557 




1 


243286 


25 


1 


2579181 24 


i 1 


272074 


23 


1 


285782 


22 


2 


243534 


50 


2 


258158 


48 


! 2 


272306 


46 


2 


286007 


45 


3 


243782 


74 


3 


258398 


72 


3 


272538 


70 


3 


286232 


67 


4 


244030 


99 


4 


258637 


96 


4 


272776 


93 


4 


286456 


89 


5 


244277 


124 


5 


258877 


120 


5 


273001 


116 


5 


286681 


112 


6 


244524 


149 


6 


259116 


144 


6 


273233 


139 


6 


286905 


134 


7 


244772 


174 


7 


259355 


167 




273464 


162 


7 


287130 


157 


8 


245019 


198 


8 


259594 


192 


8 


273696 


186 


8 


287354 


179 


^ 


245266 


222 


9 


259833 


215 


9 


273927 


209 


9 


287578 


202 


1760 


245513 




1820 


260071 i 


1880 


274158 




1940 


287802 




1 


245759 


25 


1 


260310 


24 


1 


274389 


23 


1 


288025 


22 


2 


246006 


49 


2 


260548 


48 


2 


274620 


46 


2 


288249 


45 


3 


246252 


74 


3 


260787 


71 


3 


274850 


69 


3 


288473 


67 


4 


246499 


98 


4 


261025 


95 


4 


275081 


92 


4 


288696 


89 1 


5 


246745 


123 


5 


261263 


119 


5 


275311 


115 


5 


288920 


112 


« 


246991 


148 


6 


261501 


143 


6 


275542 


138 


6 


289143 


184 


.7 


.247236 


173 


7' 


261738 


167 


7 


275772 


161 


7 


289366 


156 


8 


247482 


197 


8 


261976 


191 


8 


276002 


184 


8 


289589 


178 


9 


247728 


221 


9 


262214 


214 


9 


276232 


207 


9 


289812 


201 


1770 


247S73 




1830 


262451 




1890 


276462 




1950 


290035 




1 


248219 


25 


1 


262688 


24 


1 


276691 


23 


1 


290257 


22 j 


2 


248464 


49 


2 


262925 


47 


2 


276921 


46 


2 


290480 


44 1 


3 


248709 


74 


3 


263162 


71 


3 


277151 


69 


3 


290702 


67 1 


4 


248954 


98 


4 


263399 


95 


4 


277380 


92 


4 


290925 


89 i 


5 


249198 


123 


5 


263636 


118 


5 


277609 


115 


5 


291147 


111 1 


6 


-249443 


147 


6 


26387S 


142 


6 


277838 


138 


6 


291369 


133 j 


7 


249687 


172 


7 


264109 


166 


7 


278067 


161 


7 


291591 


156 ! 


8 


249932 


196 


8 


264345 


190 


^ 8 


278296 


183 


8 j 291813 


178! 


9 


250176 


220 


9 


264582 


213 


9 


278525 


206 


9 1 292034 


200 1 



LOGARITHMS OF NUMBERS. 



I — 

No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


^°^- ,^: 


1960 


292256 




2020 


305351 




2080 


318063 




2140 


330414 j 


1 


292478 


22 


1 


305566 


21 


1 


318272 


21 


-1 


330617 1 20 


2 


292699 


44 


2 


305781 


43 


2 


318481 


42 


2 


330819 40 


3 


292920 


66 


3 


305996 


64 


3 


318689 


63 


3 


331022 61 


4 


293141 


88 


4 


306211 


86 


4 


318898 


83 


4 


331225 1 81 


5 


293363 


110 


5 


306425 


107 


5 


319106 


104 


5 


331427 i 101 


6 


293583 


133 


6 


306639 


129 


6 


319314 


125 


6 


331630 1 121 


7 


293804 


155 


7 


306854 


150 


7 


319522 


146 


7 


331832 


141 


8 


294025 


177 


8 


307068 


172 


8 


319730 


167 


; s 


332034 


162 


9 


294246 


199 


9 


307282 


193 


9 


319938 


188 


9 


332236 


182 


1970 


294466 




2030 


307496 




2090 


320146 




,2150 


332438 




1 


294687 


22 


1 


307710 


21 


1 


320354 


21 


1 


332640 


20 


2 


294907 


44 


2 


307924 


43 


2 


320562 


41 


2 


332842 


40 


3 


295127 


66 


3 


308137 


64 


3 


320769 


62 


3 338044 


60 


4 


295347 


88 


4 


308351 


85 


• 4 


320977 


83 


4 


333246 


81 


6 


295567 


110 


5 


308564 


107 


5 


321184 


104 


5 


333447 


101 


6 


295787 


132 


6 


308778 


128 


6 


321391 


125 


6 


333649 


121 


7 


296007 


154 


7 


308991 


149 


7 


321598 


145 


i 7 


3^3850 


141 


8 


296226 


176 


8 


309204 


171 


8 


321805 


166 


8 


334051 


161 


9 


296446 


198 


9 


309417 


192 


9 


322012 


187 


9 


334253 


181 


1980 


296665 




2040 


309630 




2100 


322219 




2160 334454 




1 


296884 


22 


1 


309843 


21 


1 


322426 


21 


1 334655 


20 


2 


297104 


44 


2 


310056 


43 


2 


322633 


41 


2 334856 


40 


3 


297323 


66 


3 


310268 


64 


3 


322839 


62 


3 i 335056 


60 


4 


297542 


88 


4 


310481 


85 


4 


323046 


82 


4 j 335257 


80 


5 


297761 


109 


5 


310693 


106 


6 


323252 


103 


5 i 335458 


100 


6 


297979 


131 


6 


310906 


127 


6 


323458 


124 


6 i 335658 


120 


7 


298198 


153 


7 


311118 


148 


7 


323665 


144 


i 7 


335859 


140 


8 


298416 


175 


8 


311330 170 


8 


323871 


165 


i ^ 


336059 '. 160 


9 


298635 


197 


9 


311542 1 191 


9 


324077 


186 


9 


336260 1 180 


1990 


298853 




2050 


311754} 


2110 


324282 




2170 


336460 




1 


299071 


22 


1 


311966 21 


1 


324488 


21 


1 


336660 


20 


2 


299289 


44 


2 


312177 


42 


2 


324694 


41 


2 


336860 


40 


3 


299507 


65 


3 


312389 


63 


3 


324899 


62 


3 


337060 


60 


4 


299725 


87 


4 


312600 


84 


4 


325105 


82 


4 


337260 


80 


5 


299943 


109 


5 


312812 


106 


5 


325310 


103 


5 


337459 


100 


6 


300160 


131 


6 


313023 


127 


6 


326516 


123 


6 


337659 


120 


7 


300378 


153 


7 


313234 


148 


7 


325721 


144 


7 1 337858 


140 


8 


300595 


174 


8 


313445 


160 


8 


325926 


164 


8 


338058 1 160 


9 


300813 


196 


9 


313656 


190 


9 


326131 


185 


9 


338257 180 


2000 


301030 




2060 


313867 




2120 


326336 




,2180 


338456 




1 


301247 


22 


1 


314078 


21 


1 


326541 


20 


1 


338656 


20 


2 


301464 


43 


2 


314289 


42 


2 


326745 


41 


2 


338855 


40 


3 


301681 


65 


3 


314499 


63 


3 


326950 


61 


3 


339054 


60 


4 


301898 


87 


4 


314710 


84 


4 


327155 


82 


4 


339253 


80 


5 


302114 


108 


5 


314920 


105 


5 


327359 


102 


5 


339451 1 100 1 


6 


302331 


130 


6 


315130 


126 


6 


327563 


123 


6 


339650 


119 


7 


302547 


152 


7 


315340 


147 


7 


327767 


143 


7 


339849 


139 


8 


302764 


173 


8 


315550 


168 


8 


327972 


164 


8 


340047 


159 


9 


302980 


195 


9 


315760 


189 


9 


328176 


184 


9 


340246 


179 


2010 


303196 




2070 


315970 




2130 


328380 




2190 


340444 




1 


303412 


22 


1 


316180 


21 


1 


328583 


20 


1 


340642 


20 


2 


303628 


43 


2 


316390 


42 


2 


328787 


41 


2 


340841 


40 


3 


303844 


65 


3 


316599 


63 


3 


328991 


61 


3 
4 


341039 


59 


4 


304059 


86 


4 


316809 


84 


4 


329194 


81 


341237 


79 


5 


304275 


108 


5 


317018 


105 


5 


329398 


102 


5 


341435 


99 


6 


304490 


129 


6 317227 


126 


6 


329601 


122 


6 


341632 


119 


7 


304706 


151 


7 317436 


147 


7 


329805 


142 


7 


341830 


139 


8 


304921 


172 


8 317645 


168 


8 


330008 


163 


8 


342028 


158 


9 


305136 


194 


9 317854 


189 


9 330211 


183 


9 


342225 j 178 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


a- 


No. 


Log. 


Prop. 
Part. 


j No. 


Log. 


Prop. 
Part. 


No. 

2380 


Log. 

376577 


Prop. 
Part. 


2200 


342423 




2260 


354108 




;2320 


365488 




1 


342620 


20 


1 


354301 


19 


1 


365675 


19 


1 


376759 


18 


2 


342817 


39 


2 


354493 


38 


2 


365862 


37 


2 


376942 


36 


3 


343014 


59 


3 


354685 


58 


3 


366049 


56 


3 


377124 


55 


4 


343212 


79 


4 


354876 


77 


4 


366236 


75 


4 


377306 


73 


5 


343409 


99 


5 


355068 


96 


■5 


366423 


93 


5 


377488 


91 


6 


343606 


118 


6 


355260 


115 


6 


366610 


112 


6 


377670 


109 


7 


343802 


138 


7 


355452 


134 


7 


366796 


131 


7 


377852 


127 


8 


343999 


158 


8 


355643 


154 


8 


366983 


150 


8 


378034 


146 


9 


344196 


178 


1 9 


355834 


173 


9 


367169 


168 


9 


378216 


164 


2210 


344392 




2270 


356026 




2330 


367356 




2390 


378398 




1 


344589 


20 


1 


356217 


19 


1 


367542 


19 


1 


378580 


18 


2 


344785 


39 


2 


356408 


38 


2 


367729 


37 


2 


378761 


36 


3 


344981 


59 


3 


356599 


57 


3 


367915 


56 


3 


378943 


55 


4 


345178 


78 


4 


356790 


76 


4 


368101 


75 


4 


379124 


73 


5 


345374 


98 


5 


356981 


95 


5 


368287 


93 


5 


379306 


91 


6 


345570 


118 


6 


357172 


115 


6 


368473 


112 


6 


379487 


109 


7 


345766 


137 


7 


357363 


134 


7 


368659 


130 


7 


379668 


127 


8 


345962 


157 


8 


357554 


153 


8 


368844 


149 


8 


379849 


146 


9 


346157 


176 


9 


357744 


172 


9 


369030 


167 


9 


380030 


164 


2220 


346353 




2280 


357935 




2340 


369216 




2400 


380211 




1 


346549 


19 


1 


358125 


19 


1 


369401 


19 


1 


380392 


18 


2 


346744 


39 


2 


358316 


38 


2 


369587 


37 


2 


380573 


36 


3 


346939 


58 


3 


358506 


57 


3 


369772 


56 


3 


380754 


55 


4 


347135 


78 


4 


358696 


76 


4 


369958 


74 


4 


380934 


73 


5 


347330 


97 


5 


358886 


95 


5 


370143 


93 


5 


381115 


91 


6 


347525 


117 


6 


359076 


114 


6 


370328 


111 


6 


381296 


109 


7 


347720 


137 


7 


359266 


133 


7 


370513 


130 


7 


381476 


127 


8 


347915 


156 


8 


359456 


152 


8 


370698 


148 


8 


381656 


145 


9 


348110 


175 


9 


359646 


171 


9 


370883 


167 


9 381837 


163 


2230 


848305 




2290 


359835 




2350 


371068 




2410 382017 




1 


8-18500 


19 


1 


360025 


19 


1 


371253 


18 


1 382197 


18 


2 


848694 i 39 


2 


360215 


38 


2 


371437 


37 


2 


382377 


36 


3 


848889 


58 


a 


360404 


57 


3 


371622 


55 


3 


382557 


54 


4 


349083 


78 


4 


360593 


76 


4 


371806 


74 


4 


382737 


72 


5 


349278 


97 


5 


360783 


95 


5 


371991 


92 


5 


382917 


90 


6 


349472 


117 


6 


360972 


114 


6 


372175 


111 


6 


383097 


108 


7 


349666 


137 


7 


361161 


133 


7 


372360 


129 


7 


383277 


126 


8 


349860 


156 


8 


361350 


152 


8 


372544 


148 


8 


383456 


144 


9 


350054 


175 


9 


361539 


171 


9 


372728 


166 


9 


383636 


162 


2240 


350248 




2300 


361728 




2360 


372912 




2420 


383815 




1 


350442 


19 


1 


361917 


19 


1 


373096 


18 


1 


383995 


18 


2 


350636 


39 


2 


362105 


38 


2 


373280 


37 


2 


384174 


36 


3 


350829 


58 


3 


362294 


56 


3 


373464 


55 


3 


384353 


54 


4 


351023 


77 


4 


362482 


75 


4 


373647 


74 


4 


384533 


72 


5 


351216 


97 


5 


362671 


94 


5 


373831 


92 


5 


384712 


90 


6 


351410 


116 


6 


362859 


113 


6 


374015 


110 


6 


384891 


108 


7 


351603 


135 


7 


363048 


132 


7 


374198 


129 


7 


385070 


126 


8 I 351796 


155 


8 


363236 


151 


8 


374382 


147 


8 


385249 


144 


9 ! 351989 


174 


9 


363424 


170 


9 


374565 


166 


9 


385428 


162 


2250 


352182 




2310 


363612 




2370 


374748 




2430 


385606 




1 


352375 


1^ 


1 


363800 


19 


1 


374932 


18 


1 


385785 


18 


2 1 352568 


38 


2 


363988 


37 


2 


375115 


37 


2 


385964 


35 


3 ! 352761 


58 


3 


364176 


56 


3 


375298 


55 


3 


386142 


53 


4 : 352954 


77 


4 


364363 


75 


4 


375481 


73 


4 


386321 


71 


5 ' 353147 


96 


5 


364551 


94 


5 


375664 


92 


5 


386499 


89 


6 353339 


115 


6 


364739 


112 


6 


375846 


110 


6 


386677 


107 


7 353532 


134 


7 


364926 


131 


7 


376029 


128 


7 


386856 


125 


8 353724 


154 


8 


365113 


150 


8 


376212 


147 


8 


387034 


143 


9 353916 


173 


9 


365301 


169 


9 


376394 


165 


9 


387212 


161 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


IZl 


No. 


Log. 


a; 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


2440 


387390 




2500 397940 




2560 


408240 




2620 


418301 




1 


387568 


18 


1 


398114 


JJ 


1 


408410 


17 


1 


418467 


17 


2 


387746 


36 


2 


398287 


2 


408579 


34 


2 


418633 


33 


3 


387923 


53 


3 


398461 


53 


3 


408749 


51 


3 


418798 


50 


4 


388101 


71 


4 


398634 


69 


4 


408918 


68 


4 


418964 


66 


5 


388279 


89 


5, 


398808 


87 


5 


409087 


85 


5 


419129 


83 


6 


388456 


107 


6 


398981 


104 


6 


409257 


102 


6 


419295 


99 


7 


388634 


125 


7 


399154 


1211 


7 


409426 


119 


7 


419460 


116 


8 


388811 


142 


8 


399327 


138 


8 


409595 


1361 


8 


419625 


132 


9 


388989 


160 


9 


399501 


156; 


9 


409764 


153 j 


9 


419791 


149 


2450 


389166 




2510 


399674 


i 


2570 


409933 




2630 


419956 




1 


389343 


18 


1 


399847 


17 I 


1 


410102 


17| 


1 


420121 


16 


2 


389520 


36 


2 


400020 


35 


2 


410271 


34 


2 


420286 


33 


3 


389697 


53 


3 


400192 


53, 


3 


410440 


50 


3 


420451 


49 


4 


389875 


71 


. 4 


400365 


69! 


4 


410608 


67 


4 


420616 


66 


5 


390051 


89 


5 


400538 


87! 


5 


410777 


84 


5 


420781 


82 


6 


390228 


107 


6 


400711 


104! 


6 


410946 


1011 


6 


420945 


99 


7 


390405 


125 


7 


400883 


121' 


7 


411114 


118' 


7 


421110 


115 


8 


390582 


142 


8 


401056 


138' 


8 


411283 


135; 


8 


421275 


132 


9 


390759 


160 


9 


401228 


156! 


9 


411451 


152 


9 


421439 


148 


2460 


390935 




2520 


401400 


! 


2580 


411620 


2640 


421604 




1 


391112 


18 


1 


401573 


17 


1 


411788 


17 


1 


421768 


16 


2 


391288 


35 


2 


401745 


34 


2 


411956 


34 


2 


421933 


33 


3 


391464 


53 


3 


401917 


52 


3 


412124 


50 ! 


3 


422097 


49 


4 


391641 


70 


4 


402089 


69 


4 


412292 


67-i 


4 


422261 


66 


5 


391817 


88 


5 


402261 1 86: 


5 


412460 


84 


5 


422426 


82 


6 


391993 


106 


6 


402433 ! 103 


6 


412628 


101 1 


6 


422590 


99 


7 


392169 


123 


7 


402605 


120 


7 


412796 


1181 


7 


422754 


115 


8 


392345 


141 


8 


402777 


138 


8 


412964 


135 


8 


422918 


132 


9 


392521 


158 


9 


402949 


155 


9 


413132 


152 


9, 


423082 


148 


2470 


392697 




2530 


403120 




2590 


413300 




2650 


423246 




1 


392873 


18 


1 


403292 


17 


1 


413467 


17 


1 


423410 


16 


2 


393048 


35 


2 


403464 


34 


2 


413635 


33 


2 


423573 


33 


3 


393224 


53 


3 


403635 


52 


3 


413802 


50 


3 


423737 


49 


4 


393400 


70 


4 


403807 


69 


4 


413970 


67 


4 


423001 


65 


5 


393575 


88 


5 


403978 


86 


5 


414137 


84 


5 


424064 


81 


6 


393751 


106 


6 


404149 


103 


6 


414305 


101 


6 


424228 


98 


7 


393926 


123 


7 


404320 


120 


7 


414472 


117 


7 


424392 


114 


8 


394101 


141 


8 


404492 


137 


8 


414639 


134 


8 


424555 


131 


9 


394276 


158 


9 


404663 


154 


9 


414806 


151 


9 


424718 


147 


2480^ 


394452 




2540 


404834 




2600 


414973 




2660 


424882 


1 


1 


394627 


17 


1 


405005 


17 


1 


415140 


17 


1 


425045 


16 


2 


394802 


35 


2 


405175 


34 


2 


415307 


33 


2 


425208 


33 


3 


394977 


53 


3 


405346 


51 


3 


415474 


50 


3 


425371 


49 


4 


395162 


70 


4 


405517 


68 


4 


415641 


67 


t 


425534 


65 


5 


395326 


87 


5 


405688 


85 


5 


415808 


84 


425697 


81 


6 


395501 


104 


1 6 


405858 


102 


6 


415974 


101 


6 


425860 


98 


7 


395676 


122 


! 7 


406029 


119 


7 


416141 


117 


7 


426023 


114 


8 


395850 


139 


8 


406199 


136 


8 


416308 


134 


8 


426186 


ISO 


9 


396025 


157 


9 


406370 


153 


9 


416474 


150 


9 


426349 


147 


2490 


396199 




2550 


406540 




2610 


416640 




2670 


426511 




1 


396374 


17 


1 


406710 


17 


1 


416807 


17 


1 


426674 


16 


2 


396548 


35 


2 


' 406881 


34 


2 


416973 


33 


2 


426836 


33 


3 


396722 


53 


3 


407051 


51 


3 


417139 


50 


3 


426999 


49 


4 


396896 


70 


4 


! 407221 


68 


4 


417306 


66 


1 4 


427161 


65 


6 


397070 


87 


5 


i 407391 


85 


5 


417472 


83 


5 


427324 


81 


6 


397245 


104 


6 


' 407561 


102 


6 


417638 


100 


6 


427486 


98 


7 


397418 


122 


7 


' 407731 


119 


7 


417804 


116 


7 


427648 


114 


8 


397592 


139 


8 


407900 


136 


8 


417970 


133 


8 


427811 


130 


9 


397766 


157 


9 408070 '153 


9 


418135 1 149 


9 


427973 


147 



10 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 

Pare. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Pan. 


2680 


428135 




2740 


437751 




2800 


447158 




2860 


456366 




1 


428297 


16 


1 


437909 


16 


1 


447313 


15 


1 


456518 


15 


2 


428459 


32 


2 


438067 


32 


2 


447468 


31 


2 


456670 


30 


3 


428621 


48 


3 


438226 


47 


3 


447623 


46 


3 


456821 


46 


4 


428782 


05 


4 


438384 


63 


4 


447778 


62 


4 


456973 


61 


5 


428944 


81 


5 


438542 


79 


5 


447933 


77 


5 


457125 


76 


6 


429106 


97 


6 


438700 


95 


6 


448088 


93 


6 


457276 


91 


7 


429268 


113 


7 


438859 


111 


7 


448242 


108 


7 


457428 


106 


8 


429429 


129 


8 


439017 


127 


8 


448397 


124 


8 


457579 


122 


9 


429591 


145 


9 


439175 


143 


9 


448552 


139 


9 


457730 


137 


2690 


429752 




2750 


439333 




2810 


448706 




2870 


457882 




1 


429914 


16 


1 


439491 


16 


1 


448861 


15 


1 


458033 


15 


2 


430075 


32 


2 


439648 


32 


2 


449015 


31 


2 


458184 


30 


3 


430236 


48 


3 


439806 


47 


3 


449170 


46 


3 


458336 


45 


4 


430398 


65 


4 


439964 


63 


4 


449324 


€2 


4 


458487 


61 


5 


430559 


81 


5 


440122 


79 


5 


449478 


77 


5 458638 


76 


6 


430720 


97 


6 


440279 


95 


6 


449633 


92 


6 


458789 


91 




430881 


113 


7 


440437 


111 


7 


449787 


108 


7 


458940 


106 


8 


431042 


129 


8 


440594 


126 


8 


449941 


123 


8 


459091 


121 


9 


431203 


145 


9 


440752 


142 


9 


450095 


139 


9 


459242 


136 


2700 


431364 




2760 


440909 




2820 


450249 




2880 


459392 




1 


431525 


16 


1 


441066 


16 


1 


450403 


15 


1 


459543 


15 


2 


431685 


32 


2 


441224 


31 


2 


450557 


31 


2 


459694 


30 


3 


431846 


48 


3 


441381 


47 


3 


450711 


46 


3 


459845 


45 


4 


432007 


64 


4 


44153b 


63 


4 


450865 


62 


4 


459995 


61 


6 


432167 


80 


5 


441695 


78 


5 


451018 


77 


5 


460146 


76 


6 


432328 


96 


6 


441852 


94 


6 


451172 


92 


6 


460296 


91 


7 


432488 


112 


7 


442009 


110 


7 


451326 


108 


7 


460447 


106 


8 


432649 


128 


8 


442166 


126 


8 


451479 


123 


8 


460597 


121 


9 


432809 


144 


9 


442323 


141 


9 


451633 


139 


9 


460747 


136 


2710 


432969 




2770 


442480 




2830 


451786 




2890 


460898 




1 


433129 


16 


1 


442636 


16 


1 


451940 


15 


1 


461048 


15 


2 


433290 


32 


2 


442793 


31 


2 


452093 


31 


2 


461198 


30 


3 


433450 


48 


3 


442950 


47 


3 


452247 


46 


3 


461348 


45 


4 


433610 


64 


4 


443106 


63 


4 


452400 


61 


4 


461498 


60 


5 


433770 


80 


5 


443263 


78 


5 


452553 


77 


5 


461649 


75 


6 


433930 


96 


6 


443419 


94 


6 


452706 


92 


6 


461799 


90 


7 


434090 


112 


7 


443576 


110 


7 


452859 


107 


7 


461948 


105 


8 


434249 


128 


8 


443732 


126 


8 


453012 


123 


8 


462098 


120 


9 


434409 


144 


9 


443888 


141 


9 


453165 


138 


9 


462248 


135 


2720 


434569 




2780 


444045 




2840 


453318 




2900 


462398 




1 


434728 


16 


1 


444201 


16 


1 


453471 


15 


1 


462548 


15 


2 


434888 


32 


2 


444357 


31 


2 


453624 


31 


2 


462697 


30 


3 


435048 


48 


3 


444513 


47 


3 


453777 


46 


3 


462847 


45 


4 


435207 


64 


4 


444669 


62 


4 


453930 


61 


4 


462997 


60 


5 


435366 


80 


5 


444825 


78 


5 


454082 


77 


5 


463146 


75 


6 


435526 


96 


6 


444981 


94 


6 


454235 


92 


6 


463296 


90 


7 


435685 


112 


7 


445137 


109 


7 


454387 


107 


7 


463445 


105 


8 


435844 


128 


8 


445293 


125 


8 


454540 


123 


8 


463594 


120 


9 


436003 


144 


9 


445448 


140 


9 


454692 


138 


9 


463744 


135 


2730 


436163 




2790 


445604 




2850 


454845 




2910 


463893 




1 


436322 


16 


1 


445760 


16 


1 


454997 


15 


1 


464042 


15 


2 


436481 


32 


2 


445915 


31 


2 


455149 


30 


2 


464191 


30 


3 


436640 


47 


3 


446071 


47 


3 


455302 


46 


3 


464340 


45 


4 


436798 


63 


4 


446226 


62 


4 


455454 


61 


4 


464489 


60 


5 


436957 


79 


5 


446882 


78 


5 


455606 


76 


5 


464639 


75 


6 


437116 


95 


6 


446537 


94 


6 


455758 


91 


6 


464787 


90 


7 


437275 


111 


7 


446692 


109 


7 


455910 


106 


7 


464936 


105 


8 


437433 


127 


8 


446848 


125 


i 8 


456062 


122 


8 


465085 


120 


9 


437592 


143 


9 


447003 


140 


1 9 


456214 


137 


9 


465234 


135 



LOGARITHMS OF NUMBERS. 



11 



No. 


Log. 


Prop. 
Part. 


No. 


^os. Is.- 


No. 


^o«- ^^JlT. 


No. 


^«- 1& 


2^20 


465383 




2980 


474216 




3040 i 482874 j 




3100 


491362 


1 


1 


465532 


15 


1 


474362 


15 


1 


483016 


14 


1 


491502 


14 


2 


465680 


30 


2 


474508 


29 


2 


483159 


28 


2 


491642 


28 1 


3 


465829 


44 


3 


474653 


44 


3 


483302 


43 


3 


491782 


42 


4 


465977 


59 


4 


474799 


58 


^ 


483445 


57 


4 


491922 


56 


5 


466126 


74 


5 


474944 


73 


H 


483587 


71 


5 


492062 


70 


6 


466274 


89 


6 


475090 


88 


^ 


483730 


85 


6 


492201 


84 


7 


466423 


104 


7 


475235 


102 


7| 


483872 


99 


7 


492341 


98 


8 


466571 


118 


8 


475381 


117 


8 1 484015 ! 114 1 


8 


492481 


112 


9 


466719 


133 


9 


475526 


131 


9 1 484157 


128 


9 492621 


126 


2930 


466868 




2990 


475671 


1 


3050 1 484300 




3110 


492760 




1 


467016 


15 


1 


475816 


15 


1 


484442 


14 


1 


492900 


14 


2 


467164 


30 


2 


475962 


29 


2 


484584 


28 


2 I 493040 


28 


3 


467312 


44 


3 


476107 


43 


3 


484727 


43 


3 1493179 


42 


4 


467460 


59 


4 


476262 


58 


4 


484869 


57 


4 493319 


56 


5 


467608 


74 


5 


476397 


72 


5 


485011 


71 


5 i 493458 


70 


6 


467756 


89 


6 


476542 


87 


6 


485153 


85 


6 1 493597 


84 i 


7 


467904 


104 


7 


476687 


101 


7 


485295 


9y 


7 1 493737 ! 98 | 


8 


468052 


118 


8 


476832 


116 


8 


485437 


114 


8 


493876 i 142 


9 


468200 


133 


9 


476976 


130 1 

i 


9 


485579 


128 


9 


494015 126 


2940 


468347 




3000 


477121 


3060 


485721 




3120 


494155 




1 


468495 


15 


1 


477266 


14 1 


1 


485863 


14 


1 


494294 


14 


2 


468643 


30 


2 


477411 


29 


2 


486005 


28 


2 


494433 


28 


3 468790 


44 


3 


477555 


43 


3 


486147 


43! 


3 


494572 


41 


4 i 868938 


59 


4 


477700 


58 1 


4 


486289 


57 1 


4! 494711 


56 


5 1 469085 


74 


5 


477844 


72 


5 


486430 


71 


5 494850 


69 


6 1 469233 


89 


6 


477989 


87' 


6 


486572 


85 


6 1 494989 


83 


7 ! 469380 


104 


7 


478133 


101 


7 


486714 


99 


7 


495128 


97 


8 


469527 


118 


8 


478278 


116 


8 


486855 


114 


8 


495267 


111 


9 


469675 


133 


9 


478422 


130 


9 


486997 


128 


9 


495406 


125 


2950 


469822 




13010 


478566 




3070 


487138 




3130 


495544 




1 


469969 


15 


1 


478711 


14 


1 


487280 


14 


1 


495683 


14 


2 


470116 


29 


2 


478855 


29 


2 


487421 


28 


2 


495822 


28 


3 


470263 


44 


3 


478999 


43 


3 


487563 


42 


3 


495960 


41 


4 


470410 


59 


4 


479143 


58 


4 


487704 


57 


4 


496099 


56 


5 


470557 


74 


5 


479287 


72 


5 


487845 


71 


5 


496237 


69 


6 


470704 


88 


6 


479431 


86 


6 


487986 


85 


6 


496376 


83 


7 


470851 


103 


7 


479575 


101 


7 


488127 


99 


7 


496514 


97 


8 


470998 


118 


8 


479719 


115 


8 


488269 


113 


8 


496653 


111 


9 


471145 


132 


9 


479863 


130 


9 


488410 


127 


9 


496791 


125 


2960 


471292 




3020 


480007 




3080 


488551 




3140 


496930 




1 


471438 


15 


1 


480151 


14 


1 


488692 


14 


1 


497068 


14 


2 


471585 


29 


2 


480294 


29 


2 


488833 


28 


2 


497206 


28 


3 


471732 


44 


3 


480438 


43 


3 


488973 


42 


3 


497344 


41 


4 


471878 


59 


4 


480582 


58 


4 


489114 


56 


4 


497482 


55 


5 


472025 


73 


5 


480725 


72 


5 


489255 


70 


5 


497621 


69 


6 


472171 


88 


6 


480869 


86 


6 


489396 


84 


6 


497759 


83 


7 


472317 


102 


7 


481012 


101 


7 


489537 


98 


7 


497897 


97 


8 


472464 


117 


8 


1481156 


115 


8 


489677 


112 


8 


498035 


110 


9 


472610 


132 


9 


481299 


130 


9 


489818 


120 


9 


498173 


124 


2970 


472756 




3030 


481443 


1 


3090 


489958 




3150 


498311 




1 


472903 


15 


1 


481586 


14 


1 


490099 


14 


1 


498448 


1 14 


2 


473049 


29 


2 


481729 


29 


2 


490239 


28 


2 


498586 


j 28 


3 


473195 


44 


3 


481872 


43 


3 


490380 


42 


3 


498724 


i 41 


4 


473341 


59 


4 


482016 


57 


4 


490520 


56 


4 


498862 


1 55 


5 


1 473487 


73 


5 


482159 


71 


5 


490661 


70 


5 


498999 ; 69 1 


6 


473633 


88 


6 


482302 


86 


6 


490801 


84 


6] 499137, 83 


7 


; 473779 


102 


7 


482445 


100 


7 


1 490941 


98 


7 1 499275 97 


8 


' 473925 


117 


8 


482588 


114 


8 


491081 


112 


81499412 110 


9 


474070 


132 


9 


482731 


129 


9 '491222 


126 


9 1499550 124 

1 



12 



LOGARITHMS OF NUMBERS. 



No. 


Lo. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


^0.- \^: 


No. 


Log. 


Prop. 
Part. 


3160 


499687 




3220 


507856 




3280 


615874 




3340 


523746 




1 i 499824 


14 


1 


507991 


13 


1 


516006 


13 


1 


523876 


13 


2 


499962 


27 


2 


508125 


27 


2 


516139 


26 


2 


624006 


26 


3 


500099 


41 


3 


508260 


40 


3 


516271 


40 


3 


524136 


39 


4 


500236 


55 


4 


508395 


64 


4 


516403 


53 


4 


524266 


52 


5 


500374 


68 


5 


508530 


67 


5 


516535 


66 


5 


524396 


65 


6 


500511 


82 


6 


508664 


81 


6 


516668 


79 


6 


524526 


78 


7 


500648 


96 


7 


608799 


94 


7 


516800 


92 


7 


624656 


91 


8 


500785 


110 


8 


508933 


108 


8 


516932 


106 


8 


524785 


104 


9 


500922 


123 


9 


509068 


121 


9 


617064 


119 


9 


524915 


117 


3170 


501059 




3230 


509202 




3290 


517196 




3350 


525046 




1 


501196 


14 


1 


609337 


13 


1 


617328 


13 


1 


625174 


13 


2 


501333 


27 


2 


509471 


27 


2 


617460 


26 


2 


525304 


26 


3 


501470 


41 


3 


509606 


40 


3 


517692 


40 


3 


525434 


39 


4 


501607 


55 


4 


609740 


64 


4 


517724 


53 


4 


525563 


52 


5 


501744 


68 


5 


609874 


67 


5 


617855 


66 


5 


626693 


65 


6 


501880 


82 


6 


510008 


81 


6 


517987 


79 


6 


525822 


78 


7 


502017 


96 


7 


510143 


94 


7 


518119 


92 


7 


525951 


91 


8 


502154 


110 


8 


510277 


108 


8 


518251 


106 


8 


526081 


104 


9 


502290 


123 


9 


510411 


121 


9 


518382 


119 


9 


526210 


117 


3180 


502427 




3240 


510545 




3300 1 518514 




3360 


626339 




1 


502564 


14 


1 


510679 


13 


1 


518645 


13 


1 


626468 


13 


2 


502700 


27 


2 


510813 


27 


2 


518777 


26 


2 


526698 


26 


3 


502837 


41 


3 


510947 


40 


. 3 


518909 


39 


3 


626727 


39 


4 


502973 


54 


4 


511081 


64 


4 


619040 


52 


4 


526856 


■ 52 


6 


503109 


68 


• 5 


511215 


67 


5 


619171 


66 


5 


626985 


65 


6 


503246 


82 


6 


511348 


80 


6 


519303 


79 


6 


527114 


78 


7 


503382 


95 


7 


511482 


94 


7 


519434 


92 


7 


527243 


91 


8 


503518 


109 


8 


511616 


107 


8 


619565 


105 


8 


527372 


104 


9 


503654 


123 


9 


611750 


121 


9 


519697 


118 


9 


527601 


117 


3190 


503791 




3250 


511883 




3310 


519828 




3370 


527630 




1 


503927 


14 


1 


512017 


13 


1 


519959 


13 


1 


627759 


13 


2 


504063 


27 


2 


512160 


27 


2 


620090 


26 


2 


527888 


26 


3 


504299 


41 


3 


512284 


40 


3 


620221 


39 


3 


628016 


38 


4 


504335 


54 


4 


512417 


63 


4 


620362 


52 


4 


628145 


51 


5 


504471 


68 


5 


612551 


67 


6 


520483 


66 


6 


528274 


64 


6 


504607 


82; 


6 


512684 


80 i 


6 


620614 


79 


6 


528402 


77 


7 


504743 


95 


7 


512818 


93; 


7 


620746 


92 


7 


528531 


90 


8 


504878 


109! 


8 


512961 


107 


8 


520876 


105 


8 


528660 


103 


9 


505014 


122 


9 


513084 


120 


9 


521007 


118 


9 


628788 


116 


3200 


505150 




3260 


513218 




3320 


621138 




3380 


528917 




1 


605286 


14 1 


1 


513361 


13 


1 


521269 


13 


1 


629045 


13 


2 


505421 


27 


2 


513484 


27 1 


2 


521400 


26 


2 


529174 


26 


3 


505557 


41 


3 


613617 


40 j 


3 


521530 


39 


3 


529302 


38 


4 


505692 


54 


4 


513750 


63 


4 


521661 


52 


4 


529430 


51 


5 


505828 


68 


5 


613883 


66 


6 


521792 


65 


5 


629559 


64 


6 


505963 


82 


6 


514016 


80 


6 


521922 


78 


6 


529687 


77 


7 


506099 


95 


7 


514149 


93 


7 


522053 


97 


7 


529815 


90 


8 


506234 


109 


8 


514282 


106 


8 


522183 


104 


8 


629943 


103 


9 


506370 


122 


9 


514415 


120 


9 


522314 


117 


9 


530072 


116 


3210 


506505 




3270 


514648 




3330 


622444 




3390 


630200 




1 


506640 


13 


1 


514680 


13 


1 


522575 


13 


1 


630328 


13 


2 


506775 


27 


2 


614813 


27 


2 


622705 


26 


2 


630456 


26 


3 


506911 


40 


3 


514946 


40 


3 


522835 


39 


3 


530584 


38 


4 


507046 


54 


4 


516079 


63 


4 


522966 


52 


4 


530712 


51 


5 


507181 


67 


5 


615211 


66 


5 


523096 


65 


5 


530840 


64 


6 


507316 


81 


6 


515344 


80 


6 


523226 


78 


6 


530968 


77 


7 


507451 


94 


7 


515476 


93 


7 


523356 


97 


7' 


'531095 


90 


8 


507586 


108 


8 615609 


106 


8 


523486 


104 


8 


531223 


102 


9 507721 


121 


9 515741 


120 


9 


623616 


117 1 


9 


531351 


115 



LOGARITHMS OF NUMBERS. 



13 



No. 


Log. 


Prop. 
Part. 


No. 1 Log. 


Prop. 
Part., 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


^a^^; 


3400 


531479 


3460 539076 


' 


3520 


54G543 




3580 


553883 




1 


531607 


13 


1 ; 539202 


13 


1 


546666 


12 


1 


554004 


12 


2 


531734 


25 


2 i 539327 


25 


2 


546789 


25 


2 


554126 


24 


3 


531862 


38 


3 539452 


38 


3 


546913 


37 


3 


554247 


36 


4 


531990 


51 


4 


539578 


50 


4 


547036 


49 


4 


554368 


49 


5 


532117 


63 


5 


539703 


63 


5 


547159 


62 


5 


554489 


61 


6 


532245 


76 


6 


539829 


75 


6 


547282 


74 


6 


554610 


73 


7 


532372 


89 


7 


539954 


88 


7 


547405 


86 


7 


554731 


85 


8 


532500 


102 


8 


540079 


100 


8 


547529 


99 


8 


554852 


97 


9 


532627 


114 


9 


540204 


113 


9 


547652 


111 


9 


554973 


109 


3410 


532754 




3470 


540329 




3530 


547775 




3590 


555094 




1 


532882 


13 


1 


540455 


12 


1 


547898 


12 


1 


555215 


12 


2 


533009 


25 


2 


540580 


25 


2 


548021 


25 


2 


555336 


24 


3 


533136 


38 


3 


540705 


37 


3 


548144 


37 


3 


555457 


36 


4 


533263 


51 


4 


540830 


50 


4 


548266 


49 


4 


555578 


48 


5 


533391 


63 


5 


540955 


62 


5 


548389 


61 


5 


555699 


60 


6 


533518 


76 


6 


541080 


75 


6 


548512 


74 


6 


555820 


72 


7 


533645 


89 


7 


541205 


87 


7 


548635 


86 


7 


555940 


84 


8 


533772 


102 


8 


541330 


100 


8 


548758 


98 


8 


556061 


96 


9 


533899 


114 


9 


541454 


112 


9 


548881 


HI 


9 


556182 


108 


3420 


534026 




3480 


541579 




3540 


549003 




3600 


556302 




1 


534153 


13 


1 


541704 


12 


1 


549126 


12 


1 


556423 


12 


2 


534280 


25 


2 


541829 


25 1 


2 


549249 


25 


2 


556544 


24 


3 


534407 


38 


3 


541953 


37 


3 


549371 


37 


3 


556664 


36 


4 


534534 


51 


4 


542078 


50 


4 


549494 


49 


4 


556785 


48 


5 


534661 


63 


5 


542203 


62 


5 


549616 


61 


5 


556905 


60 


6 


534787 


76 


6 


542327 


75 


6 


549739 


74 


6 


55702G 


72 


7 


534914 


89 


7 


542452 


87 


7 


549861 


86 


7 


557146 


84 


8 


535041 


102 


8 


542576 


100 


8 


549984 


98 


8 


557267 


96 


9 


535167 


114 


9 


542701 


112 


9 


550106 


111 


9 


557387 


108 


3430 


535294 




3490 


542825 




3550 


550228 




3610 


557507 




1 


535421 


13 


1 


542950 


12 


1 


550351 


12 


1 


557627 


12 


2 


535547 


25 


2 


543074 


25 


2 


550473 


24 


2 


557748 


24 


3 


535674 


38 


3 


543199 


37 


3 


550595 


37 


3 


557868 


36 


4 


535800 


50 


4 


548323 


50 


4 


550717 


49 


4 


557988 


48 


5 


535927 


63 


5 


543447 


62 


5 


550840 


61 


5 


558108 


60 


6 


536053 


76 


6 


543571 


75 


6 


550962 


73 


6 


558228 


72 


7 


536179 


88 


7 


543696 


87 


7 


651084 


86 


7 


558348 


84 


8 


536306 


101 


8 


543820 


100 


8 


551206 


98 


8 


558469 


96 


9 


536432 


114 


9 


543944 


112 


9 


551328 


110 


9 


558589 


108 


3440 


536558 




3500 


544068 




3560 


551450 




3620 


558709 




1 


536685 


13 


1 


544192 


12 


1 


551572 


12 


1 


558829 


12 


2 


536811 


25 


2 


544316 


25 


2 


551694 


24 


2 


558948 


24 


3 


536937' 


38 


3 


544440 


37 


3 


551816 


37 


3 


559068 


36 


4 


537063 


50 


4 


544564 


50 


4 


551938 


49 


4 


559188 


48 


5 


537189 


63 


5 


544688 


62 


5 


552059 


61 


5 


559308 


60 


6 


537315 


76 


6 


544812 


74 


6 


552181 


73 


6 


559428 


72 


7 


537441 


88 


7 


544936 


87 


7 


552303 


86 


7 


559548 


84 


8 


537567 


101 


8 


545060 


99 


8 


552425 


98 


8 


559667 


96 


9 


537693 


114 


9 


545183 


112 


9 


552546 


110 


9 


559787 


108 


3450 


537819 




3510 


545307 




3570 


552668 




3630 


559907 




1 


537945 


13 


1 


545431 


12 


1 


552790 


12 


1 


560026 


12 


2 


538071 


25 


2 


545554 


25 


2 


552911 


24 


2 


560146 


24 


3 


538197 


38 


3 


545678 


37 


3 


553033 


36 


3 


560265 


36 


4 


538322 


50 


4 


545802 


49 


4 


553154 


49 


4 


560385 


48 


5 


538448 


63 


5 


545925 


62 


5 


553276 


61 


5 


560504 


60 


6 


538574 


76 


6 


546049 


74 


6 


553398 


73 


6 


560624 


72 


7 


538699 


88 


7 


546172 


86 


7 


553519 


85 


7 


560743 


84 


8 


538825 


101 


8 


546296 


99 


8 


553640 


97 


8 


560863 


96 


9 


538951 


114 


9 


546419 


111 


9 


553762 


109 


9 


560982 


108 



14 



LOGARITHMS OF NUMBERS. 



1 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


1 

Log. 


Prop. 
Pait. 


No. 


Log. 


Prop. 
Part. 


3640 


561101 




3700 


568202 




3760 


575188 




3820 


582063 




1 


561221 


12 


1 


568319 


12 


1 


575303 1 


12 


1 


582177 


11 


2 


561340 


24 


2 


568436 


23 


2 


575419 


23 


2 


582291 


23 


3 


561459 


36 


3 


568554 


35 


3 


575534 


36 


3 


582404 1 34 


4 


561578 


48 


4 


568671 


47 


4 


575650 


46 


4 


682518! 45 


5 


561698 


60 


5 


568788 


58 


5 


575765 


58 


5 


6826311 66 


6 


561817 


72 


6 


568905 


70 


6 


575880 


69 


6 


582746: 68 


7 


561936 


84 


7 


569023 


82 


7 


575996 


80 


7 


582858; 79 


8 


562055 


96 


8 


569140 


94 


8 


576111 


92 


8 


682972 90 


9 


562174 


108 


9 


569257 


106 


9 


576226 


104 


9 


683085 102 


3650 


562293 




3710 


569374 




3770 


676341 




3830 


583199 


1 


562412 


12 


1 


569491 


12 


1 


676457 


12 


1 


583312 11 


2 


562531 


24 


2 


569608 


23 


2 


576572 


23 


2 


583426 23 


3 


562650 


36 


3 


569725 


35 


3 


576687 


36 


3 


583539 1 34 


4 


562768 


48 


4 


569842 


47 


4 


676802 


46 


4 


583652 1 46 


5 


562887 


60 


5 


569959 


58 


5 


576917 


58 


5 


583765 1 66 


6 


563006 


71 


6 


570076 


70 


6 


577032 


69 


6 


588879, 68 


7 


563125 


83 


7 


570193 


82 


7 


577147 


80 


7 


683992: 79 


8 


563244 


95 


8 


570309 


94 


8 


577262 


92 


8 


584105; 90 


9 


563362 


107 


9 


570426 


106 


9 


577377 


104 


9 


684218 102 


3660 


563481 




3720 


570543 




3780 1 577492 




3840 


584331 1 


1 


563600 


12 


1 


570660 


12 


1 : 577607 


11 


1 


584444' 11 


. 2 


563718 


24 


2 


570776 


23 


2 577721 23 


2 


584657! 23 


3 


563837 


36 


3 


570893 


35 


3; 577836] 34 


3 


584670! 34 


4 


563955 


48 


4 


571010 


47 


4 5779511 46 


4 


684783 45 


5 


564074 


60 


5 


571126 


58 


5 578066' 57 


5 
6 


584896: 56 


6 


564192 


71 


6 


571243 


70 


6 


578181 


68 


686009 68 


7 


564311 


83 


7 


571359 


81 


7 


578295 


80 


7 


585122 79 


8 


564429 


95 


8 


571476 


93 


8 


578410 


91 


1 ^ 


585235 90 


9 


564548 


107 


9 


571592 


105 


9 


578525 


103 


! 9 


585348 102- 


3670 


564666 




3730 


571709 




3790 


578639 




3850 


685461 


1 


564784 


12 


1 


571825 


12 


1 


578764 


11 


1 


585574} 11 


2 


564903 


24 


2 


571942 


23 


2 


578868 


23 


2 


685686! 22 


3 


565021 


36 


3 


572058 


35 


3 


578983 


34 


3 


5857991 34 


4 


565139 


47 


4 


572174 


47 


4 


579097 


46 


4 


585912 1 45 


5 


565257 


59 


5 


572291 


58 


5 


579212 


57 


6 


586024! 56 


6 


565376 


71 


6 


572407 


70 


6 


579326 


68 


6 


586137 j 67 


7 


565494 


83 


7 


572523 


81 


7 


579441 


80 


7 


5862501 78 


8 


565612 


95 


8 


572639 


93 


8 


579556 


91 


8 


5863621 90 


9 


565730 


107 


9 


572755 


105 


9 


579669 


103 


9 


686475 101 


3680 


565848 




3740 


572872 




3800 


679784 




3860 


586587 


1 


565966 


12 


1 


572988 


12 


1 


579898 


11 


1 


586700 j 11 


2 


566084 


24 


2 


573104 


23 


2 


680012 


23 


2 


5868121 22 


3 


566202 


35 


3 


573220 


35 


3 


680126 


34 


3 


6869261 34 


4 


566320 


47 


4 


573336 


46 


4 


580240 


46 


4 


6870371 45 


5 


566437 


59 


5 


573452 


58 


6 


580355 


57 


6 


687149! 56 


6 


566555 


71 


6 


573568 


70 


6 


580469 


68 


6 


687262! 67 


7 


566673 


83 


7 


573684 


81 


7 


580583 


80 


7 


687374! 78 


8 


566791 


94 


8 


573800 


93 


8 


580697 


91 


8 


587486 1 90 


9 


566909 


106 


9 


573915 


104 


9 


580811 


103 


9 


■587699 


101 


3690 


567026 




3750 


574031 




3810 


580925 




3870 


587711 




1 


567144 


12 


1 


574147 


12 


1 


581039 


11 


1 


587823 


11 


2 


567262 


24 


2 


674263 


23 


2 


681153 


23 


2 


587936 


22 


3 


567379 


35 


3 


574379 


35 


3 


581267 


34 


3 


588047 


34 


4 


567497 


47 


4 


674494 


46 


4 


581381 


46 


4 


588160; 45 


5 


567614 


59 


5 


574610 


58 


5 


681495 


67 


5 


588272! 56 


6 


567732 


71 


6 


574726 


70 


6 


581608 


68 


6 


588384' 67 


7 


567849 


83 


7 


574841 


81 


7 


581722 


80 


7 


588496! 78 


8 


567967 


94 


8 


674957 


93 


! 8 


681836 


91 


8 


588608 90 


9 ' 568084 


106 


9 575072 


104 


9 


581960 


103 


9 


588720 101 



LOGARITHMS OF NUMBERS. 



15 



No. 


Log. 


Prop. 
Part. 


No. 


^0.. |i-r 


r 
No. 


Log. 


Prop. 
Part. 


No. 

4060 


Lo,. 


Prop.l 
Pari.: 


3880 


588832 




3940 


595496 




4000 


602060 




608526 




1 


588944 


11 


1 


596606 


11 


1 


602169 


11 


1 


608633 


11 


2 


589056 


22 


2 


595717 


22 


2 


602277 


22 


2 


608740 


21 


3 


589167 


33 


3i 


595827 


33 


3 


602386 


33 


. 3 608847 


32 


4 


589279 


44 


4 i 595937 1 


44 


4 


602494 


43 


4 


608954 


43 1 


5 


589391 


56 


5 1 596047 


55 


5 


602603 


54 


5 


609061 


53 


6 


589503 


67 


6 1 596157 


66 


6 


602711 


65 


6 


609167 !^64 1 


7 


589615 


78 


7 1 596267 


77 


7 


602819 


76 


7 


609274 75 1 


8 


589726 


89 


8 i 596377 


88 


8 


602928 


87' 


8 


609381 86 : 


9 


589838 


100 


9 


596487 


99 


9 


603036 


98 


9 


609488 1 96 1 


3890 


589950 




3950 


596597 




4010 


603144 




4070 


609594 1 i 


1 


590061 


11 


1 


596707 


11 


1 


603253 


11 


1 


609701 


11 


2 


590173 


22 


2 


596817 


22 


2 


603361 


22 


2 


609808 


21 


3 


590284 


33 


3 


596927 


33 


3 


603469 


33 


3 


609914 


32 i 


4 


590396 


44 


4 


597037 


44 


4 


603577 


43 


4 


610021 


43 


5 


590507 


56 


5 


597146 


55 


5 


603686 


54 


5 


610128 


53 


6 


590619 


67 


6 


597256 


66 


6 


603794 


65 


6 


610234 


64 


7 


590730 


78 


7 


597366 


77 


7 


603902 


76 


7 


610341 


75 


8 


590842 


89 


8 


597476 


88 


8 


604010 


87 


8 


610447 


86 i 


9 


590953 


100 


9 


597585 


99 


9 


604118 


98 


9 


610554 


96 1 


3900 


591065 




3960 


597695 




4020 


604226 




4080 


610660 


1 


1 


591176 


11 


1 


597805 


11 


1 


604334 


11 


1 


610767 


11 i 


2 


591287 


22 ' 


2 


597914 


22 


2 


604442 


22 


2 


610873 


21 1 


3 


591399 


33 


3 


598024 


33 


3 


604550 


32 


3 


610979 


32 ! 


4 


591510 


44 


4 


598134 


44 


4 


604658 


43 


4 


611086 


42 i 


5 


591621 


56 


5 


598243 


55 


5 


604766 


54 


5 


611192 


53 1 


6 


591732 


67 


6 


598353 


66 


6 


604874 


65 


6 


611298 


64 


7 


591843 


78 


7 


598462 


77 


7 


604982 


76 


7 


611405 


74 


8 


591955 


89 


8 


598572 


88 


8 


605089 


86 


8 


611511 


85 


9 


592066 


100 


9 


598681 


99 


9 


605197 


97 


9 


611617 


95 


3910 


592177 




3970 


598790 




4030 


605305 




4090 


611723 




1 


592288 


11 


1 


598900 


11 


1 


605413 


11 


1 


611829 


11 


2 


592399 


22 


2 


599009 


22 


2 


605521 


22 


2 


611936 


21 


3 


592510 


33 


3 


599119 


33 


3 


605628 


32 


3 


612042 


32 


4 


592621 


44 


4 


599228 


44 


4 


605736 


43 


4 


612148 


42 


5 


592732 


55 


5 


599337 


55 


5 


605844 


54 


5 


612254 


53 


6 


592843 


67 


6 


599446 


66 


6 


605951 


65 


6 


612360 


64 


7 


592954 


78 


7 


599556 


77 


7 


606059 


76 


7 


612466 


74 


8 


593064 


89 


8 


599665 


88 


8 


606166 


86 


8 


612572 


86 


9 


593175 


100 


9 


599774 


99 


9 


606274 


97 


9 


612678 


96 


3920 


593286 




3980 


599883 




4040 


606381 




4100 


612784 




1 


593397 


11 


1 


599992 


11 


1 


606489 


11 


1 


612890 


11 


- 2 


593508 


22 


2 


600101 


22 


2 


606596 


21 


2 


612996 


21 


3 


593618 


33 


3 


600210 


33 


3 


606704 


32 


3 


613101 


32 


4 


593729 


44 


4 


600319 


44 


4 


606811 


43 


4 


613207 


42 


5 


593840 


55 


5 


600428 


54 


5 


606919 


54 


5 


613313 


53 


6 


593950 


66 


6 


600537 


65 


6 


607026 


64 


6 


613419 


64 


7 1 594061 


77 


7 


600646 


76 


7 


607133 


75 


7 


613525 


74 


8 


594171 


88 


8 


600755 


87 


8 


607241 


86 


8 


613630 


85 


9 


594282 


99 


9 


600864 


95 


9 


607348 


96 


9 


613736 


95 


3930 


594393 




3990 


600973 




4050 


607455 




4110 


613842 




1 


594503 


11 


1 


601082 


11 


1 


607562 


11 


1 


613947 


11 


2 ' 594613 


22 


2 


601190 


22 


2 


607669 


21 


2 


614053 


21 


3 594724 


33 


3 


601299 


33 


3 


607777 


32 


3 


614159 


32 


4 ! 594834 


i 44 


4 


601408 


44 


4 


607884 


43 


4 


614264 


42 


5 1 594945 


! 55 


5 


601517 


54 


6 


607991 


54 


5 


614370 


53 


6 


595055 


1 66 


6 


601625 


65 


6 


608098 


64 


6 


614475 


63 


7 


595165 


77 


7 


601734 


76 


7 


608205 


75 


7 


614581 


74 


8 


595276 


88 


8 


601843 


87 


8 


608312 


86 


8 


614686 


84 i 


9 


595386 


99 


9 


601951 


98 


9 


608419 


96 


9 


614792 1 95 j 



16 



LOGARITHMS OF. NUMBERS. 



No. 

4120 


Log. 

614897 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Ifr?.- 


No. 


Log. 


Prop. 
Part. 




4180 


621176 




14240 


627366 




4300 


633468 




1 


615003 


11 


1 


621280 


10 


! 1 


627468 


10 


1 


633569 


10 


2 


615108 


21 


2 


621384 


21 


i 2 


627571 


20 


2 


633670 


20 


'3 


615213 


31 


3 


621488 


31 


3 


627673 


31 


3 


633771 


30 


4 


615319 


42 


4 


621592 


42 


4 


627775 


41 


4 


633872 


40 


5 


615424 


52 


i 5 


621695 


52 


5 


627878 


51 


5 


633973 


50 


6 


615529 


63 


6 


621799 


62 


6 


627980 


61 


6 


634074 


61 


7 


615634 


73 


7 


621903 


73 


7 


628082 


72 


7 


634175 


71 


8 


615740 


84 


8 


622007 


83 


8 


628184 


82 


8 


634276 


81 


9 


615845 


95 


9 


622110 


94 


9 


628287 


92 


9 


634376 


91 


4130 


615950 




4190 


G22214 




4250 


628389 




4310 


634477 




1 


616055 


11 


1 


622318 


10 


1 


628491 


10 


1 


634578 


10 


2 


616160 


21 


2 


622421 


21 


2 


628593 


20 


2 


634679 


20 


3 


616265 


31 


3 


622525 


31 


3 


628695 


31 


3 1 634779 


30 


4 


616370 


42 


4 


622628 


41 


4 


628797 


41 


4 


634880 


40 


5 


616475 


52 


5 


622732 


52 


5 


628900 


51 


! 5 


634981 


50 


6 


616580 


63 


6 


622835 


62 


6 


629002 


61 


1 6 


635081 


61 


7 


616685 


73 


7 


622939 


72 


7 


629104 


72 


7 


635182 


71 


8 


616790 


84 


8 


623042 


83 


8 


629206 


82 


8 


635283 


.81 


9 


616895 


95 


9 


623146 


93 


9 


629308 


92 


9 


635383 


91 


4140 


617000 




4200 


623249 




4260 


629410 




4320 


635484 




1 


617105 


10 


1 


623353 


10 


1 


629511 


10 


1 


635584 


10 


2 


617210 


21 


2 


623456 


21 


2 


629613 


20 


2 


635685 


20 


3 


617315 


31 


3 


623559 


31 


3 


629715 


80 


3 


685785 


30 


4 


617420 


42 


4 


623663 


41 


4 


629817 


41 


4 


635886 


40 


5 


617524 


52 


5 


623766 


52 


5 


629919 


51 


6 


635986 


50 


6 


617629 


63 


1 6 


623869 


62 


6 


630021 


61 


6 


636086 


60 


7 


617734 


73 


i 7 


623972 


72 


7 


630123 


71 


7 


636187 


70 


8 


617839 


84 


8 


624076 


83 


8 


630224 


81 


8 


636287 


80 


9 


617943 


94 


9 


624179 


93 


9 


630326 


91 


9 


636388 


90 


4150 


618048 




4210 


624282 




4270 


630428 




4330 ! 636488 




1 


618153 


10 


1 


624385 


10 


1 


630530 


10 


1 


636588 


10 


2 


618257 


21 


2 


624488 


21 


2 


630631 


20 


2 


636688 


20 


3 


618362 


31 


3 


624591 


31 


3 


630733 


30 


3 


636789 


30 


4 


618466 


42 


4 


624694 


41 


4 


630834 


41 


4 


636889 


40 


5 


618571 


52 


5 


624798 


51 


5 


630936 


51 


5 


636989 


50 


6 


618675 


62 


6 


624901 


62 


6 


631038 


61 i 


6 


637089 


60 


7 


618780 


73 


7 


625004 


72 


7 


631139 


71 ! 


7 


637189 


70 


8 


618884 


83 


8 


625107 


82 


8 


681241 


81 


8 


637289 


80 


9 


618989 


94 


9 


625209 


93 


9 


631312 


91 


9 


637390 


90 


4160 


619093 




4220 


625312 




4280 


631444 




4340 


637490 




1 


619198 


10 


1 


625415 


10 


1 


631545 


10 


1 


637590 


10 


2 


619302 


21 


2 


625518 


21 


2 


631647 


20 


2 


637690 


20 


3 


619406 


31 


3 


625621 


31 


3 


681748 


30 ! 


3 


637790 


30 


4 


619511 


42 


4 


625724 


41 


4 


631849 


41 


4 


637890 


40 


5 


619615 


62 


5 


625827 


51 


5 


631951 


51 


5 


637990 


50 


6 


619719 


62 


6 


625929 


62 


6 


632052 


61 


6 


638090 


60 


7 


619823 


73 


7 


626032 


72 


7 


632153 


71 


7 


638190 


70 


8 


619928 


83 


8 


626135 


82 


8 


632255 


81 


8 


638289 


80 


9 


620032 


94 


9 


626238 


93 


9 


632356 


91 


9 


638389 


90 


4170 


620186 




4230 


626340 




4290 


632457 




4350 


638489 




1 


620240 


10 


1 


626443 


10 


1 


632558 


10 


1 


638589 


10 


id 


620344 


21 


2 


626546 


21 


2 


682660 


20 


2 


638689 


20 


3 


620448 


31 


3 


626648 


31 


3 


632761 


30 


3 


638789 


30 


4 


620552 


42 


4 


626751 


41 


4 


632862 


41 


4 


638888 


40 


5 


620656 


52 


5 


626853 


51 


5 


632963 


51 


5 


638988 


50 


6 


620760 


62 


6 


626956 


62 


6 


633064 


61 


6 


639088 


60 


7 


620864 


73 


7 


627058 


72 


7 


633165 


71 


7 


639188 


70 


8 


620968 


83 


8 


627161 


82 


8 


633266 


81 


8 


639287 


80 


9 


621072 


94 


9 


627263 


93 


9 


633367 


91 


9 


639387 


90 



LOGARITHMS OF NUMBERS. 



17 



No. j Log. 


^:i 


No. 

4420 


I^og. & 


No. 

4480 


Log. 

651278 


Prop 
Part. 


'■ No. 

1 


Log. 


Prop. 
Part. 


4360 


639486 




645422 


! 




4540 


657056 




1 


639586 


10 


1 


645520 


' 10 


1 


651375 


10 


1 


657151 


10 


2 ; 639686 


20 


2 


645619 


; 20 


2 


651472 


19 


2 


657247 


19 


3 ' 639785 


30 


3 


645717 


30 


3 


651569 


29 


! 3 


657343 


28 


4 j 639885 


40 


4 


645815 


39 


4 


651666 


38 


4 


657438 


38 


5 639984 


50 


6 


645913 


49 


5 


651762 


48 


5 


657534 


47 


6 ! 640084 


60 


6 


646011 


59 


6 


651859 


58 


6 


657629 


57 


. 7 


1 640183 


70 


7 


646109 


69 


7 


651956 


67 


7 


657725 


67 


8 


640283 


80 


8 


646208 


79 


8 


652053 


77 


8 


657820 


76 


9 


640382 


90 


9 


646306 


89 


9 


652150 


87 


9 


657916 


86 


4370 


640481 




4430 


646404 




4490 


652246 




4550 


658011 




1 


940581 


10 


1 


646502 


10 


1 


652343 


10 


1 


658107 


10 


2 


640680 


20 


2 


646600 


20 


2 


652440 


19 


2 


658202 


19 


3 


640779 


30 


3 


646698 


29 


3 


1 652536 


29 


3 


658298 


28 


4 


640879 


40 


4 


646796 


39 


4 


: 6526S3 


38 


4 


658393 


38 


5 


640978 


50 


5 


646894 


49 


5 


652730 


48 


5 


658488 


47. 


6 


641077 


60 


6 


646991 


59 


6 


652826 


58 


6 


658584 


57 


7 


641176 


70 


7 


647089 


69 


7 


652923 


67 


7 


658679 


67 


8 


641276 


80 


8 


647187 


78 


8 


653019 


77 


8 


658774 


76 


9 


641375 


90 


9 


647285 


88 


9 


653116 


87 


9 


658870 


86 


4380 


641474 




4440 


647383 




4500 


653213 




4560 


658965 




1 


641573 


10 


1 


647481 


10 


1 


653309 


10 


1 


659000 


10 


2 


641672 


20 


2 


647579 


20 


2 


653405 


19 


2 


659155 


19 


3 


641771 


30 


3 


647676 


29 


3 


653502 


29 


3 


659250 


28 


4 


641870 


40 


4 


647774 


39 


4 


653598 


38 


• 4 


659346 


38 


5 


641970 


50 


5 


647872 


49 


6 


653695 


48 


5 


659441 


47 


6 


642069 


59 


6 


647969 


59 ' 


6 


653791 


58 


6 


659536 


57 


7 


642168 


69 


7 


648067 


69 ' 


7 


653888 


67 


7 


659631 


67 


8 


642267 


79 


8 


648165 


78 


8 


653984 


77 


8 


659726 


76 


9 


642366 


89 


9 


648262 


88 i 


9 


654080 


87 


9 


659821 


86 


4390 


642464 




4450 


648360 




4510 


654176 




4570 


659916 




1 


642563 


10 


1 


648458 


10 1 


1 


654273 


10 


1 


660011 


10 


2 


642662 


20 


2 


648555 


19 i 


2 


654369 


19 


2 


660106 


19 


3 


642761 


30 


3 


648653 


29 j 


3 


654465 


29 


a 


660201 


28 


4 


642860 


40 


4 


648750 


39 


4 


654562 


38 


4 


660290 


38 


5 


642959 


49 


5 


648848 


49 


5 


654558 


48 


5 


660391 


47 


6 


643058 


59 


6 


648945 


68 1 


6 


654754 


58 


6 


660486 


67 


7 


643156 


69 


7 


649043 


68 1 


7 


654850 


67 


7 


660581 


67 


8 


643255 


79 


8 


649140 


78 


8 


654946 


77 


8 


660676 


76 


9 


643354 


89 


9 


649237 


88 


9 


655042 


86 


9 


660771 


86 


4400 


643453 




4460 


649335 




4520 


655138 




4580 


660865 


- 


1 


643551 


10 


11 


649432 


10 


1 


655234 


10 


1 


660960 


9 


2 


643650 


20 


2! 


649530 


19 


2 


655331 


19 


2 


661055 


19 


3 


643749 


30 


3 


649627 


29 


3 


655427 


29 


3 


661150 


28 


4 


643847 


39 


4 


649724 


39 1 


4 


655523 


38 


4 


661245 


38 


5 


643946 


49 


5 


649821 


49 1 


5 


655619 


48 


6 


661339 


47 


6 


644044 


59 


6 


649919 


58 1 


6 


655714 


58 


6 


661434 


67 


7 


644143 


69 


7| 


650016 


68 


7 


655810 


67 


7 


661529 


66 


8 


644242 


79 


8' 


650113 


78 


8 


655906 


77 


8 


661623 


76 


9 


644340 


89 


9| 


650210 


88 


9 


656002 


86 


9 


661718 


85 


4410 


644439 




44701 


650307 




4530 


656098 


1 


4590 


661813 




1 


644537 


10 


1 


650405 


10 


1 


656194 


10 i 


1 


661907 


9 


2 


644635 


20 


21 


650502 


19 


2 


656290 


19 


2 


662002 


19 


3 


644734 


30 


3i 


650599 


29 


3 


656386 


29 


3 


662096 


28 


4 


644832 


39 


4| 


650696 


39 


4 


656481 


38 


4 


662191 


38 


6 


644931 


49 


5' 


650793 


49 


5 


656577 


48 


5 


662285 


47 


6 


645029 


59 


61 


650890- 


58 


6 


656673 


58 1 


6 


662380 


67 


7 


645127 


69 


7 i 


650987 


68 


7 


656769 


67 


7 


662474 


66 


8 


645226 


79 


8 


651084 


78 


8 


656864 


77 


8 


662569 


76 


9 


645324 


89 


9 


651181 


^8 


9 


656960 


86 


9| 


662663 


85 



18 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Par:, 


No. 


Log. 


Pfop.l 
Part. 


4600 


662758 




4660 


668386 




4720 


673942 




4780 


679428 




1 662852 1 


9 


1 


668479 


9 


1 


674034 


9 


1 


679519 


9 


2 


662947 


19 


2 


668572 


19 


2 


674126 


18 


2 


679610 


18 1 


3 


663041 


28 


3 


668665 


28 


3 


674218 


28 


3 


679700 


27 


4 


663135 


38 


4 


668758 


37 


4 


674310 


37. 


4 


679791 


36 


5 


663230 


47 


5 


668852 


47 


5 


674402 


46 


5 


679882 


45 


6 


663324 


57 


6 


668945 


56 


6 


674494 


55 


6 


679973 


55 


7 


663418 


66 


7 


669038 


65 


7 


674586 


64 


7 


680063 


64 


8 


663512 


76 


8 


669131 


74 


8 


674677 


74 


8 


680154 


73 


9 


663607 


S5 


9 


669224 


84 


9 


674769 


83 


9 


680245 


82 


4610 


663701 




4670 


669317 




4730 


674861 




4790 


680335 




1 


663795 


9 


1 


669410 


9 


1 


674953 


9 


1 


680426 


9 


2 


663889 


19 


2 


669503 


19 


2 


675045 


18 


2 


680517 


18 


3 


663983 


28 


3 


669596 


28 


3 


675136 


28 


3 


680607 


27 1 


4 


664078 


38 


4 


669689 


37 


4 


675228 


37 


4 


680698 


36 1 


5 


664172 


47 


5 


669782 


47 


5 


675320 


46 


5 


680789 


45 


6 


664266 


56 


6 


669875 


56 


6 


675412 


55 


6 


680879 


55 


7 


664360 


66 


7 


669967 


65 


7 


675503 


64 


7 


680970 


64 


8 


664454 


75 


8 


670060 


74 


8 


675595 


74 


8 


681060 


73 


9 


664548 


85 


9 


670153 


84 


9 


675687 


83 


9 


681151 


82 


4620 


664642 




4680 


670246 




4740 


675778 




4800 


681241 




1 


664736 


9 


1 


670339 


9 


1 


675870 


9 


1 


681332 


9 


2 


664830 


19 


2 


670431 


18 


2 


675962 


18 


2 


681422 


18 


3 


664924 


28 


3 


670524 


28 


3 


676053 


27 


3 


681513 


27 


4 


665018 


S9 


4 


670617 


37 


4 


676145 


36 


4 


681603 


36 


5 


665112 


47 


5 


670710 


46 


5 


676236 


46 


5 


681693 


45 


6 


665206 


56 


6 


670802 


55 


6 


676328 


55 


6 


681784 


54 


7 


665299 


66 


7 


67U8U5 


64 


7 


676419 


64 


7 


681874 


63 


8 


665393 


75 


8 


670y»8 


74 


8 


676511 


73 


8 


681964 


72 


9 


665487 


85 


9 


671080 


83 


9 


676602 


82 


9 


682055 


81 1 


4630 


665581 




4690 


671173 




4750 


676694 




4810 


682145 




1 


665675 


9 


1 


671265 


9 


1 


676785 


9 


1 


682235 


9 


2 


665769 


19 


2 


671358 


18 


2 


676876 


18 


2 


682326 


18 


3 


665862 


28 


3 


671451 


28 


3 


676968 


27 


3 


682416 


27 


4 


665956 


38 


4 


671543 


37 


4 


677059 


36 


4 


682506 


36 


5 


666050 


47 


5 


671636 


46 


5 


677151 


46 


5 


682596 


45 i 


6 


666143 


56 


6 


671728 


55 





677242 


55 


6 


682686 


54 


7 


666237 


66 


7 


671821 


64 


7 


677333 


64 


7 


682777 


63 


8 


666331 


75 


8 


671913 


74 


8 


677424 


73 


8 


682867 


72 


9 


666424 


85 


9 


672005 


83 


9 


677516 


82 


9 


682957 


81 


4640 


666518 




4700 


672098 




4760 


677607 




4820 


683047 




1 


666612 


9 


1 


672190 


9 


1 


677698 


9 


1 


683137 


9 


2 


666705 


19 


2 


672283 


18 


2 


677789 


18 


2 


683227 


18 


3 


666799 


28 


3 


672375 


28 


3 


677881 


27 


3 


683317 


27 


4 


666892 


37 


4 


672467 


37 


4 


677972 


36 


4 


683407 


36 


5 


666986 


47 


5 


672560 


46 


5 


678063 


45 


5 


683497 


45 


6 


667079 


56 


6 


672652 


55 


6 


678154 


55 


6 


683587 


54 


7 


667173 


65 


7 


672744 


64 


7 


678245 


64 


7 


683677 


63 


8 


667266 


74 


8 


672836 


74 


8 


678336 


73 


8 


683767 


72 


9 


607359 


84 


9 


672929 


83 


9 


678427 


82 


9 


683857 


81 


4650 


667453 




4710 


673021 




4770 


678518 




4830 


683947 




1 


i 667546 


9 


1 


673113 


9 


1 


678609 


9 


1 


684037 


9 


2 


i 667640 


19 


2 


673205 


18 


2 


678700 


18 


2 


684127 


18 


3 


! 667733 


28 


3 


673297 


28 


3 


678791 


27 


3 


684217 


27 


4 


! 667826 


37 


4 


673390 


37 


4 


678882 


36 


4 


684307 


36 


5 


! 667920 


47 


5 


673482 


46 


5 


678973 


45 


5 


684396 


45 


6 


668013 


56 


6 


673574 


55 


6. 


679064 


55 


6 


684486 


54 


7 


668106 


65 


7 


673666 


64 


! 7 


679155 


64 


7 


684576 


63 


8 


668199 


74 


8 


673758 


74 


! 8 


679246 


73 


8 


684666 


72 


i ' 


668293 


84 


9 


673850 


83 


9 


679337 


82 


9 


684756 


81 



LOGARITHMS OF NUxMBERS. 



19 



No. 


Log. 


X'ro, 


No. 


'^ i& 


No. j Lug. ! 


Prop. 1 
Pare. 


No. 


Log. 


S| 


4840 


084845 




4900 


i 
690196 


4960 695482 




5020 


700704 


1 


084935 


9 


1 


690285 


9 


1 


695569 


9 


1 


700790 


9 1 


2 


085025 


18 


2 


690373 


18 


2 


695657 


17 


2 


700877 


17 ! 


3 


685114 


27 


3 


690462 


27 


3 


695744 


26 


3 700963 


26 


4 


685204 


36 


4 


690550 


35 


4 


695832 


35 


4 


701050 


35 


6 


685294 


45 


5 


690639 


44 


5 


695919 


44 


5 


701136 


43 1 


6 


685383 


54 


6 


690727 


53 


6 


696007 


52 


6 


701222 


52 1 


7 


685473 


63 


7 


690816 


62 


7 


696094 


61 


y 


701309 


61 


8 


685563 


72 


8 


690905 


71 


8 


696182 


70 


8 


701395 


70 


,9 


685652 


81 


9 


690993 


80 


9 


696269 


79 


9 


701482 


78 


4850 


685742 




4910 


691081 




4970 


696356 




5030 


701568 




1 


6»5831 


9 


1 


691170 


9 


1 


696444 


9 


1 


701654 


9 


o 


685921 


18 


2 


691258 


18 


2 


696531 


17 


2 


701741 


17 


3 


686010 


27 


3 


691347 


27 


3 


696618 


26 


3 


701827 


2b* 


4 


686100 


36 


4 


691435 


35 


4 


696706 


35 


4 


701913 


35 


5 


686189 


45 


5 


691524 


44 


5 


696793 


44 


5 


701999 


43 


6 


686279 


54 


6 


69161:^ 


53 


6 


696880 


52 


6 


702086 


52 


7 


686368 


63 


7 


691700 


62 


7 


696968 


61 


7 


702172 


61 


8 


686457 


72 


8 


691789 


71 


8 


697055 


70 


8 


702258 


70 


9 


686547 


81 


9 


691877 


80 


9 


697142 


79 


9 


702344 


78 1 


4860 


686636 




4920 


691965 




4980 


697229 




5040 


702430 




1 


686726 


9 


1 


692053 


9 


1 


697317 


9 


1 


702517 


9 


2 


686815 


18 


2 


692142 


18 


2 


697404 


17 


2 


702603 


17 


1 3 


686904 


27 


3 


692230 


27 


3 


697491 


26 


3 


702689 


26 


4 


686994 


36 


4 


692318 


35 


4 


697578 


35 


4 


702775 


34 


5 


687083 


45 


5 


69:^406 


44 


5 


697665 


44 


» 5 


702861 


43 


« 


687172 


54 


6 


692494 


53 


6 


697752 


52 


6 


702947 


52 


1 _ 


687261 


63 


7 


692583 


62 


7 


697839 


61 


7 


703033 


60 


8 


687351 


72 


8 


692671 


71. 


8 


697926 


70 


8 


703119 


69 


9 


687440 


81 


9 


692759 


80 


9 


698013 


79 


9 


703205 


77 


4870 


687529 




4930 


692847 




4990 


698100 




5050 


703291 




1 


687618 


9 


1 


692935 


9 


1 


698188 


9 


1 


703377 


9 


2 


687707 


18 


2 


693023 


18 


2 


698275 


17 


2 


703463 
703549 


17 


3 


687796 


27 


3 


693111 


26 


3 


698362 


26 


3 


26 


4 


687886 


36 


, 4 


693199 


35 


4 


698448 


35 


4 


703635 


34 


5 


687975 


45 


i ^ 


693287 


44 


5 


698535 


44 


5 


703721 


43 


6 


688064 


54 


1 6 


693375 


53 


6 


698622 


62 


6 


703807 


52 


7 


688153 


62 


1 7 


693463 


62 


7 


698709 


61 


7 


703893 


60 


8 


688242 


72 


8 


693551 


70 


8 


698796 


70 


8 


703979 


69 


9 


688331 


80 


9 


693639 


79 


9 


698883 


79 


9 


704066 


77 


4880 


688420 




4940 


693727 




5000 


698970 




5060 


704150 




1 


688509 


9 


1 


693815 


9 


1 


699057 


9 


1 


704236 


9 


2 


688598 


18 


2 


693903 


18 


2 ; 699144 


17 


2 


704322 


17 


3 


688687 


27 


3 


693991 


26 


3 , 699231 


26 


3 


704408 


26 


4 


688776 


36 


4 


694078 


35 


4 i 699317 


35 


4 


704494 


34 


5 


688865 


45 


5 


694166 


44 


5 


699404 


43 


5 


704579 


43 


6 


688953 


54 


6 


694254 


53 


6 


699491 


52 


6 


704665 


52 


7 


689042 


62 


7 


694342 


62 


7 


699578 


61 


7 


704751 


60 


8 


689131 


72 


8 


694430 


70 


8 


699664 


70 


8 


704837 


69 


9 


689220 


80 


9 


694517 


79 


9 699751 


78 


9 


704922 


77 


4890 


689309 




4950 


694605 




5010 1 699838 




5070 


705008 




1 


689398 


9 


1 


6946% 


9 


1 


699924 


9 


1 j 705094 


9 


2 


689486 


18 


2 


694781 


18 


2 


700011 


17 


21705179 


17 


3 


689575 


27 


3 


694868 


26 


3 


700098 


20 


3 j 705265 


26 


4 


689664 


36 


4 


694956 


35 


4 


700184 


35 


4 705350 


34 


5 


689753 


45 


5 


695044 


1 44 


5 


700271 


43 


5 i 705436 


43 


6 


689841 


54 


6 


695131 


1 53 


6 


700358 


52 


6 1 705522 


62 


7 


68H930 


62 


7 


695219 


62 


7 1 700444 


61 


7 


705607 


60 


8 


690019 


72 


8 


695307 


70 


8 j 700531 


70 


8 


705693 


69 


'^ 


690107 


80 


9 


1 tj9')394 


79 


9 1 700617 


78 


9 


705778 


77 



33 



20 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


^^:| 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


I'^l: 


No. 


Log. 


Prop. 
Part. 


5080 


705864 


! 
9I 


5140 


710963 




5200 


716003 




5260 


720986 




1 


705949 


1 


711048 


8 


1 


716087 


8 


1 


721068 


8 


2 


706035 


17 1 


2 


711132 


17 


2 


716170 


17 


2 


721151 


16 


3 


706120 


26 ' 


3 


711216 


25 


3 


716254 


25 


3 


721233 


26 


4 


706206 


34 ' 


4 


711301 


34 


4 


716337 


34 


4 


721316 


33 


5 


706291 


43 1 


5 


711385 


42 


5 


716421 


42 


5 


721398 


41 


6 


706376 


51 1 


6 


711470 


51 


6 


716504 


50 


6 


721481 


49 


7 


706462 


60 


7 


711554 


59 


7 


716588 


59 


7 


721563 


58 


8 


706547 


68 I 


8 


711638 


68 


8 


716671 


67 


8 


721646 


66 


9 


706632 


77 1 


9 


711723 


76 


9 


716754 


76 


9 


721728 


74 


5090 


706718 


1 


5150 


711807 




5210 


716838 




5270 


721811 




1 


706803 


9 1 


1 


711892 


8 ; 


1 


716921 


8 


1 


721893 


8 


2 


706888 


17 1 


2 


711976 


17 i 


2 


717004 


17 


2 


721975 


16 


3 


706974 


26 1 


3 


712060 


25 1 


3 


717088 


25 


3 


722058 


25 


4 


707059 


34 1 


4 


712144 


34 i 


4 


717171 


33 


4 


722140 


33 


5 


707144 


43 ■• 


5 


712229 


42 


5 


717254 


42 


5 


722222 


41 


6 


707229 


51 1 


6 


712313 


51 1 


6 


717338 


50 


6 


722305 


49 


7 


707315 


60 


7 


712397 


59 i 


7 


717421 


58 


7 


722387 


58 


■ 8 


707400 


68 


8 


712481 


68 1 


8 


717504 


66 


8 


722469 


66 


9 


707485 


77 


9 


712566 


76 j 


9 


717587 


75 


9 


722552 


74 


5100 


707570 


i 


5160 


712650 




5220 


717671 




5280 


722634 




1 


707655 


9 \ 


1 


712734 


8 


1 


717754 


8 


1 


722716 


8 


2 


707740 


17 i 


2 


712818 


17 


2 


717837 


17 


2 


722798 


16 


3 


707826 


26 


3 


712902 


25 


3 


717920 


25 


3 


722881 


25 


4 


707911 


34 


4 


712986 


34 


4 


718003 


33 


4 


722963 


33 


5 


707996 


4.3.! 


5 


713070 


42 


5 


718086 


42 


5 


723045 


41 


6 


708081 


51 


6 


713154 


50 


6 


718169 


50 


6 


723127 


49 


7 


708166 


60 


7 


713238 


59 1 


7 


718253 


68 


7 


723209 


58 


8 


708251 


68 ■ 


8 


713322 


67 ! 


8 


718336 


66 


8 


723291 


66 


9 


758336 


77 


9 


713406 


76 


9 


718419 


75 


9 


723374 


74 


5110 


708421 




5170 


713490 




5230 


718502 




5290 


723466 




1 


708506 


9 i 


1 


713574 


8 


1 


718585 


8 


1 


723638 


8 


2 


708591 


17 1 


2 


713658 


17 


2 


718668 


17 


2 


723620 


16 


3 


708676 


26 


3 


713742 


25 


3 


718751 


25 


3 


723702 


25 


4 708761 


34 


4 


713826 


34 


4 


718834 


33 


4 


723784 


33 


5 708846 


43 


5 


713910 


42 


5 


718917 


42 


5 


723866 


41 


6 , 708931 


51 


6 


713994 


50 


6 


719000 


50 


6 


723948 


49 


7 


709015 


GO 


7 


714078 


59 


7 


719083 


58 


7 


724030 


57 


8 


709100 


68 


8 


714162 


67 


8 


719165 


66 


8 


724112 


66 


9 


709185 


77 


9 


714246 


76 


9 


719248 


75 


9 


724194 


74 


5120 


709270 




5180 


714330 




5240 


719331 




5300 


724276 




1 


709355 


8 


1 


714414 


8 


1 


719414 


8 


1 


724358 


8 


2 


709440 


17 


2 


714497 


17 


2 


719497 


17 


2 


724440 


16 


3 


709524 


25 


3 


714581 


25 


3 


719580 


25 


3 


724522 


25 


4 


709609 


34 


4 


714665 


34 


4 


719663 


33 


4 


724603 


33 


5 


709694 


42 


5 


714749 


42 


5 


719745 


41 


5 


724685 


41 


6 


709779 


51 


6 


714832 


50 


6 


719828 


50 


6 


724767 


49 


7 


709863 


59 


7 


714916 


59 


7 


719911 


58 


7 


724849 


67 


8 709948 


68 


8 


715000 


67 


8 


719994 


66 


8 


724931 


66 


9 710033 


76 


9 


716084 


76 


9 


720077 


16 


9 


725013 


74 


5130 710117 




5190 


715167 




5250 


720159 




5310 


725095 




1 : 710202 


8 


1 


715251 


8 


1 


720242 


8 


1 


725176 


8 


2 ' 710287 


17 


2 


715335 


17 


2 


720325 


17 


2 


725258 


16 


3 710371 


25 


3 


715418 


25 


3 


720407 


25 


3 


725340 


25 


4 710456 


34 


4 


715502 


34 


4 


720490 


33 


4 


725422 


33 


5 ; 710540 


42 


5 


715586 


42 


6 


720573 


41 


5 


725503 


41 


6 : 710625 


51 


6 


715669 


50 


^ 6 


720655 


50 


6 


725585 


49 


7 710710 


59 


7 


715753 


59 


7 


720738 


58 


7 


725667 


57 


8 710794 


67 


8 


715836 


67 


8 


720821 


66 


8 


725748 


66 


9 710879 


76 


9 


715920 


76 


9 


720903 


75 


9 


725830 


74 



LOGARITHMS OF NUMBERS. 



21 



No. 


Log. 


Prop. 
Part. 


No. 


I'og. ,?-?: 


No. 


Log. 


Prop 
Part. 


NO. 


Log. 


Prop. 
Part, j 


6320 


725912 




5380 


730782 




5440 


735599 




5500 


740363 


1 


1 


725993 


8 


1 


730863 


8 


1 


735679 


8 


1 


740442 


8 


2 


726075 


16 


2 


730944 


16 


2 


735759 


16 


2 


740521 


16 


3 


726156 


24 


3 


731024 


24 


3 


735838 


24 


3 


740599 


24 


4 1 726238 


33 


4 


731105 


32 


4 


735918 


32 


i 4 


740678 


32 


6 


726320 


41 


5 


731186 


40 


5 


735998 


40 


5 


740757 40 


6 


726401 


49 


6 


731266 


49 


6 


736078 


48 


6 


740836 47 


7 


726483 


57 


7 


731347 


57 


7 


736157 


56 


7 


740915 55 ! 


8 


726564 


65 


8 


731428 


65 


8 


736237 


64 


8 


740994 


63 


9 


726646 


73 


9 


731508 


73 


9 


736317 


72 


9 


741073 


71 


5330 


726727 




5390 


731589 




5450 


736396 




5510 


741152 




1 


726809 


8 


1 


731669 


8 


1 


736476 


8 


1 


741230 


8 


2 


726890 


16 


2 


731750 


16 


2 


736556 


16 


2 


741309 


16 


3 


726972 


24 


3 


731830 


24 


3 


736635 


24 


3 


741388 


24 


4 


727053 


33 


4 


731911 


32 


4 


736715 


32 


4 


741467 


32 


5 


727134 


41 


5 


731991 


40 


5 


736795 


40 


5 


741546 


40 


6 


727216 


49 


6 


732072 


48 


6 


736874 


48 


6 


741624 


47 


7 


727297 


57 


7 


7321.52 


56 


7 


736954 


56 


7 


741703 


55 


8 


727379 


65 


8 


732233 


64 


8 


737034 


64 


8 


741782 


63 


9 


727460 


73 


9 


732313 


72 


9 


737113 


72 


9 


741860 


71 


5340 


727541 




5400 


732394 




5460 


737193 




5520 


741939 




1 


727623 


8 


1 


732474 


8 


1 


737272 


8 


1 


742018 


8 


2 


727704 


16 


2 


732555 


16 


2 


737352 


16 


2 


742096 


16 


3 


727785 


24 


3 


732635 


24 


3 


737431 


24 


3 


742175 


23 


4 


727866 


33 


4 


732715 


32 


4 


737511 


32 


4 


742254 


31 


5 


727948 


41 


5 


732796 


40 


5 


737590 


40 


5 


742332 


39 


6 


728029 


49 


6 


732876 


48 


6 


737670 


48 


6 


742411 


47 


7 


728110 


57 


7 


732956 


56 


7 


737749 


56 


7 


742489 


55 


8 


728191 


65 


8 


733037 


64 


8 


737829 


64 


8 


742568 


63 


9 


728273 


73 


9 


733117 


72 


9 


737908 


72 


9 


742647 


71 


5350 


728354 




5410 


733197 




5470 


737987 




5530 


742725 




1 


728435 


8 


1' 


733278 


8 


1 


738067 


8 


1 


742804 


8 


2 


728516 


16 


2 


733358 


16 i 


2 


738146 


16 


2 


742882 


16 


3 


728597 


24 


3 


733438 


24 I 


3 


738225 


24 


3 


742961 


23 


4 


728678 


33 i 


4 


733518 


32 


4 


738305 


32 


4 


743039 


31 


5 


7287.59 


41 1 


5 


733598 


40 1 


5 


738384 


40 


5 


743118 


39 


6 


728841 


49 


6 


733679 


48 \ 


6 


738463 


48 


6 


743196 


47 


■ 7 


728922 


57 


7 


733759 


56 


7 


738543 


56 


7 


743275 


55 


8 


729003 


65 


8 


733839 


64 


8 


738622 


64 


8 


743353 


63 


9 


729084 


73 


9 


733919 


72 


9 


738701 


72 


9 


743431 


71 


5360 


729165 


1 


5420 


733999 




5480 


738781 




6540 


743510 




1 


729246 


8 ! 


1 


734079 


8 


1 


738860 


8 


1 


743588 


8 


2 


729327 


16 1 


2 


734159 


16 


2 


738939 


16 


2 


743667 


16 


3 


729408 


24 


3 


734240 


24 


3 


739018 


24 


3 


743745 


23 


4 


729489 


32 


4 


734320 


32 


4 


739097 


32 


4 


743823 


31 


5 


729570 


41 


5 


734400 


40 


5 


739177 


40 


5 


743902 


39 


6 


729651 


49 


6 


734480 


48 


6 


739256 


47 


6 


743980 


47 


7 


729732 


57 


7 


734560 


56 


7 


739335 


55 


7 


744058 


55 


8 


729813 


65 


8 


734640 


64 


8 


739414 


63 


8 


744136 


63 


9 


729893 


73 


9 


734720 


72 


9 


739493 


71 


9 


744215 


71 


5370 


729974 




5430 


734800 




5490 


739572 




6550 


744293 




1 


730055 


8 


1 


734880 


8 


1 


739651 


8 


1 


744371 


8 


2 


730136 


16 


2 


734960 


16 


2 


739730 


16 


2 


744449 


16 


3 


730217 


24 


3 


735040 


24 


8 


739810 


24 


3 


744528 


23 


4 


730298 


32 


4 


735120 


32 


4 


739889 


32 


4 


744606 


31 


5 


730378 


40 


5 


735200 


40 


5 


739968 


40 1 


5 


744684 


39 


6 


730459 


49 


6 


735279 


48 


6 


740047 


47 ! 


6 


744762 


47 


7 


730540 


57 


7 


735359 


56 


7 


740126 


55 


7 


744840 


55 


8 


730621 


65 


8 


735439 


64 


8 


740205 


63 


8 


744919 


63 


9 


730702 


73 


9 


735519 


72 


9 


740284 


71 ' 


^ 


744997 


71 



22 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


^0^- ^^:\ 


5560 


745075 




5620 


749736 




5680 


754348 




5740 


758912 




1 


745153 


8 


1 


749814 


8 


1 


754425 


8 


1 


758988 


8 


2 


745231 


16 


2 


749891 


16 


2 


754501 


15 


2 


759063 


15 


3 


745309 


23 


3 


749968 


23 


3 


754578 


23 


3 


759139 


23 


4 


745387 


31 


4 


750045 


31 


4 


754654 


30 


4 


759214 


30 


5 


745465 


39 


5 


750123 


39 


5 


754730 


38 


5 


759290 


38 


6 


745543 


47 


6 


750200 


47 


6 


754807 


46 


6 


759366 


45 


7 


745621 


55 


7 


750277 


54 


7 


754883 


53 


7 


759441 


53 


8 


745699 


62 


8 


750354 


62 


8 754960 


61 


8 


759517 


60 


9 


745777 


70 


9 


750431 


70 


9 


755036 


69 


9 


759592 


68 


5570 


745855 




5630 


750508 




5690 


755112 




5750 


759668 


1 


1 


745933 


8 


1 


750586 


8 


1 


755189 


8 


1 


759743 


8 


2 


746011 


16 


2 


750663 


16 


2 


755265 


15 


2 


759819 


15 


3 


746089 


23 


3 


750740 


23 


3 


755341 


23 


3 


759894 


23 


4 


746167 


31 


4 


750817 


31 


4 


755417 


30 


4 


759970 


30 


5 


746245 


39 


5 


750894 


39 


5 


755494 


38 


5 


760045 


38 


6 


746323 


47 


6 


750971 


47 


6 


755570 


46 


6 


760121 


45 


7 


746401 


55 


7 


751048 


54 


7 


755646 


53 


7 


760196 


53 


8 


746479 


62 


8 


751125 


62 


8 


755722 


61 


8 


760272 


60 


9 


746556 


70 


9 


751202 


70 


9 


755799 


69 


9 


760347 


68 


5580 


746634 




5640 


751279 




5700 


755875 




5760 


760422 




1 


746712 


8 


1 


751356 


8 


1 


755951 


8 


1 


760498 


8 


2 


746790 


16 


2 


751433 


15 


o 


756027 


15 


2 


760573 


15 


3 


746868 


23 


3 


751510 


23 


3 


756103 


23 


3 


760649 


23 


4 


746945 


31 


4 


751587 


30 


4 


756180 


30 


4 


760724 


30 


5 


747023 


39 


5 


751664 


88 


5 


756256 


38 


5 


760799 


38 


6 


747101 


47 


6 


751741 


46 


6 


756332 


46 


6 


760875 


45 


7 


747179 


55 


7 


751818 


54 


7 


756408 


53 


7 1 760950 


53 


8 


747256 


62 


8 


751895 


62 


8 


756484 


61 


8 1 761025 


60 


9 


747334 


70 


9 


751972 


70 


9 


756560 


69 


9 ! 761100 


68 


5590 


747412 




5650 


752048 




5710 


756636 




5770 1761176 




1 


747489 


8 


1 


752125 


8 


1 


756712 


8 


1 


761251 


8 


2 


747567 


16 


2 


752202 


15 


2 


756788 


15 


2 


761326 


15 


3 


747645 


23 


3 


752279 


23 


3 


756864 


23 


3 


761402 


23 


4 


747722 


31 


4 


752356 


30 


4 


756940 


30 


4 


761477 


30 


5 


747800 


39 


5 


752433 


38 


5 


757016 


38 


5 


761552 


38 


6 


747878 


47 


6 


752509 


46 


6 


757092 


46 


6 


761627 


45 


7 


747955 


54 


7 


752586 


54 


7 


757168 


53 


7 


761702 


53 


8 


748033 


62' 


8 


752663 


62 


8 


757244 


61 


8 


761778 


60 


9 


748110 


70 


9 


752740 


70 


9 


757320 


69 


9 


761853 


68 


5600 


748188 




5660 


752816 




5720 


757396 




5780 


761928 




1 


748266 


8 


1 


752893 


8 


1 


757472 


8 


1 


762003 


8 


2 


748343 


16 


2 


752970 


15 


2 


757548 


15 


2 


763078 


16 


3 


748421 


23 


3 


753047 


23 


3 


757624 


23 


3 


762153 


22 


4 


748498 


31 


4 


753123 


30 


4 


757700 


30 


4 


762228 


30 


5 


748576 


39 


5 


753200 


38 


5 


757775 


38 


5 


762303 


38 


6 


748653 


47 


6 


753277 


46 


6 


757851 


46 


6 


762378 


45 


7 


748731 


54 


7 


753353 


54 


7 


757927 


53 


7 


762453 


52 


8 


748808 


62 


8 


753430 


62 


8 


758003 


61 


8 


762529 


60 


9 


748885 


70 


9 


753506 


70 


9 


758079 


68 


9 


762604 


68 


i 5610 


748963 




5670 


753583 




5730 


758155 




5790 


762679 




1 


749040 


8 


1 


753660 


8 


1 


758230 


8 


1 


762754 


8 


2 


749118 


16 


2 


753736 


15 


2 


758306 


15 


2 


762829 


15 


3 


749195 


23 


3 


753813 


23 


3 


758382 


23 


3 


762904 


22 


4 


749272 


31 


4 


753889 


30 


4 


758458 


30 


4 


762978 


30 


5 


749350 


39 


5 


753966 


38 


5 


758533 


38 


5 


763053 


38 


6 


749427 


47 


6 


754042 


46 





758609 


46 


6 


763128 


45 


7 


749504 


54 


7 


754119 


54 


7 


758685 


53 


7 


763203 


52 


8 


749582 


62 


8 


754195 


62 


8 ' 758760 


61 


8 


763278 


60 


9 


749659 


70 


9 


754272 


70 


9 758836 


68 


9 


763353 


68 



LOGARITHMS OF NUMBERS. 



23 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


i-os. I|-P- 


No. 


Log. 


Prop. 
Part. 


5800 


763428 




5860 


767898 




5920 


772322 




5980 


776701 




1 


763503 


7 


1 1 767972 


7 


1 


772395 


7 


1 


776774 


7 


2 


763578 


15 


2 


768046 


15 


2 


772468 


15 


2 


776846 


14 


3 


763653 


22 


3 


768120 


22 


3 


772542 


22 


3 


776919 


22 


4 


763727 


30 


4 


768194 


30 


4 


772615 


29 


4 


776992 


29 


5 


763802 


37 


5 


768268 


37 


5 


772688 


37 


5 


777064 


36 ;■ 


6 


763877 


45 


6 


768342 


45 


6 


772762 


44 


6 


777137 


43 


7 


763952 


52 


7 


768416 


52 


7 


772835 


51 


7 


777209 


51 


8 


764027 


60 


8 


768490 


59 


8 


772908 


59 


8 


777282 


58 


9 


764101 


67 


9 


768564 


67 


9 


772981 


66 


9 


777354 


65 


5810 


764176 




5870 


768638 




5930 


773055 




6990 


777427 




1 


764251 


7 


1 


768712 


7 


1 


773128 


7 


1 


777499 


7 


2 


764326 


15 


2 


768786 


15 


2 


773201 


15 


2 


777572 


14 


3 


764400 


22 


3 


768860 


22 


3 


773274 


22 


3 


777644 


22 


4 


764475 


30 


4 


768934 


30 


4 


773348 


29 


4 


777717 


29 


5 


764550 


37 


5 


769008 


37 


•5 


773421 


37 


5 


777789 


36 


6 


764624 


45 


6 


769082 


45 


6 


773494 


44 


6 


777862 


43 


7 


764699 


52 


7 


769156 


52 


7 


773567 


51 


7 


777934 


51 


8 


764774 


60 


8 


769230 


59 


8 


773640 


59 


8 


778006 


58 


9 


764848 


67 


9 


769303 


67 


9 


773713 


66 


9 


778079 


65 


5820 


764923 




5880 


769377 




5940 


773786 




6000 


778151 




1 


764998 


7 


1 


769451 


7 


1 


773860 


7 


1 


778224 


7 


2 


765072 


15 


2 


769525 


15 


2 


773933 


15 


2 


778296 


14 


3 


765147 


22 


3 


769599 


22 


3 


774006 


22 


3 


778368 


22 


4 


765221 


30 


4 


769673 


30 


4 


774079 


29 


4 


778441 


29 


5 765296! 37 


5 


769746 


37 


5 


774152 


37 


5 


778513 


36 


6 765370 I. 45 


6 


769820 


45 


6 


774225 


44 


6 


778585 


43 


7 


765445 j 52 | 


7 


769894 


52 


7 


774298 


51 


7 


778658 51 


8 


765520 


60 


8 


769968 


59 


8 


774371 


59 


8 


778730 58 


9 


765594 


67 


9 


770042 


67 


9 


774444 


66 


9 


778802 


65 


5830 


765669 




5890 


770115 




5950 


774517 




6010 778874 




1 


765743 


7 


1 


770189 


7 


1 


774590 


7 


1 1 778947 


7 


2 


765818 


15 


2 


770263 


15 


2 


774663 


15 


2 ! 779019 


14 


3 765892 


22 


3 


770336 


22 


3 


774736 


22 


3 i 779091 


22 


4 ! 765966 


30 


4 


770410 


30 


4 


774809 


29 


4 


779163 


29 


_ 



766041 


37 


5 


770484 


37 


5 


774882 


37 


5 


779236 


36 


6 


766115 


45 


6 


770557 


45 


6 


774955 


44 


6 


779308 


43 


7 


766190 


52 


7 


770631 


52 


7 


775028 


51 


7 


779380 


51 


8 


766264 


60 


8 


770705 


59 


8 


775100 


59 


8 


779452 


58 


9 


766338 


67 


9 


770778 


67 


9 


775173 


66 


9 


779524 


65 


5840 


766413 




5900 


770852 




5960 


775246 




6020 


779596 




1 


766487 


7 


1 


770926 


7 


1 


775319 


7 


1 


779669 


7 


2 


766562 


15 


2 


770999 


15 


2 


775392 


15 


2 


779!741 


14 


3 


766636 


22 


3 


771073 


22 


3 


775465 


22 


3 


779813 


22 


4 


766710 


30 


4 


771146 


30 


4 


775538 


29 


4 1 779885 


29 


5 


766785 


37 


5 


771220 


37 


5 


775610 


37 


5 


779967 


36 


6 


766859 


45 


6 


771293 


45 


6 


775683 


44 


6 


780029 


43 


7 


766933 


52 


7 


771367 


52 


7 


775756 


51 


7 


?80101 


50 


8 


767007 


60 


8 


771440 


59 


8 


775829 


59 


81780173 


58 


9 


767082 


67 


9 


771514 


67 


9 


775902 


66 


9 1 780245 


66 


5850 


767156 




5910 


771587 




5970 


775974 




6030 


780317 




1 


767230 


7 


1 


771661 


7 


1 


776047 


7 


1 


780389 


7 


2 


767304 


15 


2 


771734 


15 


2 


776120 


15 


2 


780461 


14 


3 


767379 


22 


3 


771808 


22 


3 


776193 


22 


3 


780533 


22 


4 


767453 


30 


4 


771881 


30 


4 


776265 


29 


4 


780605 


29 


5 


767527 


37 


5 


771955 


37 


5 


776338 


37 


6 


780677 


36 


6 


767601 


45 


6 


772028 


44 


6 


776411 


44 


6 


780749 


43 


7 


767675 


52 


7 


772102 


52 


7 


776483 


51 


7 


780821 


50 


8 


767749 


59 


8 


772175 


59 


8 


776556 


59 


8 


780893 


58 


9 


767823 


67 


9 


772248 


67 


9 


776629 


66 


9 ' 780965 


.ii. 



24 



LOGARITHMS OF NUMBERS. 



No, 


Log. 


Prop. 
Part. 


No. 


Log. 


S:i.- 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 1 


6040 


781037 




6100 


785330 




6160 


789581 




6220 


793790 




1 


781109 


7 


1 


785401 


7 


1 


789651 


7 


1 


793860 


7 


2 


781181 


14 


2 


785472 


14 


2 


789722 


14 


2 


793930 


14 


3 


781253 


22 


3 


785543 


21 


3 


789792 


21 


3 


794000 


21 


4 


781324 


29 


4 


785615 


28 


4 


789803 


28 


4 


794070 


28 


5 


781396 


36 


5 


785686 


36 


5 


789933 


35 ! 


5 


794139 


35 


6 


781468 


43 


6 


785757 


43 


6 


790004 


42 


6 


794209 


42 


7 


781540 


50 


7 


785828 


50 


7 


790074 


49 


7 


794279 


49 


8 


781612 


58 


8 


785899 


57 


8 


790144 


56 


8 


794349 


56. 


9 


781684 


65 


9 


785970 


64 


9 


790215 


63 i 


9 


794418 


63 


6050 


781755 




6110 


786041 


i 
1 


6170 


790285 




6230 


794488 




1 


781827 


7 


1 


786112 


7 . 


1 


790356 


7 


1 


794558 


7 


2 


781899 


14 


2 


786183 


14 


2 


790426 


14 


2 


794627 


14 


3 


781971 


22 


3 


786254 


21 


3 


790496 


21 1 


3 


794697 


21 


4 


782042 


29 


4 


786325 


28 


4 


790567 


28 


4 


794767 


28 


5 


782114 


36 


5 


786396 


SQ 


5 


790637 


35 


5 


794836 


35 


6 


782186 


43 


6 


786467 


43 


6 


790707 


42 


6 


794906 


42 


7 


782258 


50 


7 


786538 


50 


7 


790778 


49 


7 


794976 


49 


8 


782329 


58 


8 


786609 


57 


8 


790848 


56 


8 


795045 


56 


9 


782401 


65 


9 


780680 


64 


9 


790918 


63 


9 


795115 


63 


6060 


782473 




6120 


786751 


i 


6180 


790988 




6240 


795185 




1 


782544 


7 


1 


786822 


7 S 


1 


791059 


7 


1 


795254 


7 


2 


782616 


14 


2 


786893 


14 


2 


791129 


14 


2 


795324 


14 


3 


782688 


21 


3 


786964 


21 


3 


791199 


21 


3 


795393 


21 


4 


782759 


29 


4 


787035 


28 


4 


791269 


28 


4 


795463 


28 


5 


782831 


36 


5 


787106 


36 


5 


791340 


35 


5 


795532 


35 


6 


782902' 


43 


6 


787177 


43 


6 


791410 


42 


6 


795602 


42 


7 


782974 


50 


7 


787248 


50 


7 


791480 


49 


7 


795671 


49 


8 


783046 


57 


8 


787319 


57 


8 


791550 


56 


8 


795741 


56 


9 


783117 


64 


9 


787390 


64 


9 


791620 


63 


9 


795810 


63 


6070 


783189 




6130 


787460 




6190 


791691 




6250 


795880 




1 


783260 


7 


1 


787531 


7 


1 


791761 


7 


1 


795949 


7 


2 


783332 


14 


2 


787602 


14 


2 


791831 


14 


2 


796019 


14 


3 


783403 


21 


3 


787673 


21 


3 


791901 


21 


3 


796088 


21 


4 


783475 


29 


4 


787744 


28 


4 


791971 


28 


4 


796158 


28 


5 


783546 


36 


5 


787815 


35 


5 


792041 


35 


5 


796227 


35 


6 


783618 


43 


6 


787885 


42 


6 


792111 


42 


6 


796297 


42 


7 


783689 


50 


7 


787956 


49 


7 


792181 


49 


7 


796366 


49 


8 


783761 


57 


8 


788027 


56 


8 


792252 


56 


8 


796436 


56 


9 


783832 


64 


9 


788098 


63 


9 


792322 


63 


9 


796505 


63 


6080 


783904 




6140 


788168 




6200 


792392 




16260 


796574 




1 


783975 


7 


1 


788239 


7 


1 


792462 


7 


1 


796644 


7 


2 


784046 


14 


2 


788310 


14 


9 


792532 


14 


2 


796713 


14 


3 


784118 


21 


3 


788381 


21 


3 


792602 


21 


3 


796782 


21 


4 


784189 


29 


4 


788451 


28 


4 


792672 


28 


4 


796852 


27 


6 


784261 


36 


5 


788522 


35 


6 


792742 


35 


5 


796921 


35 


6 


784332 


43 


6 


788593 


42 


6 


792812 


42 


6 


796990 


42 


7 


784403 


50 


7 


788663 


49 


7 


792882 


49 


7 


797060 


49 


8 


784475 


57 


8 


788734 


56 


8 


792952 


56 


8 


797129 


56 


9 


784546 


64 


9 


788804 


63 


9 


793022 


63 


9 


797198 


62 


6090 


784617 




6150 


788875 




6210 


793092 




6270 


797268 




1 


784689 


7 


1 


788946 


7 


1 


793162 


7 


1 


797337 


7 


2 


784760 


14 


2 


789016 


14 


2 


793231 


14 


2 


797406 


14, 


3 


784831 


21 


3 


789087 


21 


3 


793301 


21 


3 


797475 


21 


4 


784902 


29 


4 


789157 


28 


4 


793371 


28 


4 


797545 


27 


6 


784974 


36 


5 


789228 


35 


5 


793441 


35 


5 


797614 


35 


6 


785045 


43 


6 


789299 


42 


6 


793511 


42 


6 


797683 


42 


7 


785116 


60 


7 


789369 


49 


7 


793581 


49. 


7 


797752 


49 


8 


785187 


57 


8 


789440 


56 


8 


793651 


56 


8 


797821 


56 


9 


785259 


64 


9 


789510 


63 


9 


793721 


63 


9 


797890 


62 



LOGARITHMS OF NUMBERS. 



25 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


i'roj. 
Part 


6280 


797960 


i 


6340 


802089 




6400 


806180 




6460 


810233 




1 


798029 


7 j 


1 


802158 


7 


1 


806248 


7 


1 


810300 


7 


2 


798098 


14 ' 


2 


802226 


14 i 


2 


806316 


14 


2 


810367 


13 


3 


798167 


21 1 


3 


802295 


21 ! 


3 


806384 


20 


3 


810434 


20 


4 


798236 


28 1 


4 


802363 


27 1 


4 


806451 


27 


4 


810501 


27 


5 


798305 


34 


5 


802432 


34 


5 


806519 


34 


5 


810569 


33 


6 


798374 


41 i 


6 


802500 


41 1 


6 


806587 


41 


6 


810636 


40 


7 


798443 


48 i 


7 


802568 


48 ' 


7 


806655 


48 


7 


810703 


47 


8 


798512 


55 1 


8 


802637 


55 I 


8 


806723 


54 


8 


810770 


54 


9 


798582 


62 


9 


802705 


62 


9 


806790 


61 


9 


810837 


60 i 


6290 


798651 




6350 


802774 




6410 


806858 




6470 


810904 




1 


798720 


7 


1 


802842 


7 1 


1 


806926 


7 


1 


810971 


7 


2 


798789 


14 I 


2 


802910 


14 


2 1 806994 


14 


2 


811038 


13 


3 


798858 


21 


3 


802979 


21 


3 


807061 


20 


3 


811106 


20 


4 


798927 


28 


4 


803047 


27 


4 


807129 


27 


4 


811173 


27 


5 


798996 


34 ' 


5 


803116 


34 


5 


807197 


34 


5 


811240 


33 


6 


799065 


41 I 


6 


803184 


41 


6 


807264 


41 


6 


811307 


40 


7 


799134 


48 1 


7 


808252 


48 


7 


807332 


48 


7 


811374 


47 


8 


799203 


55 


8 


803820 


55 


8 


807400 


54 


i 8 


811441 


54 


9 


799272 


62 1 


9 


803389 


62 


9 


807467 


61 


9 


811508 


60 


6300 


799341 




6360 


803457 




6420 


807535 




6480 


811575 




1 


799409 


7 ' 


1 


803525 


7 


1 


807603 


7 


1 


811642 


7 j 


2 


799478 


14 , 


2 


803594 


14 


2 


807670 


14 


2 


811709 


13 


3 


799547 


21 


3 


803662 


21 


3 


807738 


20 


3 


811776 


20 ! 


4 


799616 


28 


4 


803730 


27 


4 


807806 


27 


4 


811843 


27 1 


5 


799685 


34 


5 


803798 


34 


5 


807873 


34 


5 


811910 


33 j 


6 


799754 


41 


6 


803867 


41 


6 


807941 


41 


6 


811977 


40 ! 


7 


799823 


48 


7 


803935 


48 


7 


808008 


48 


7 


812044 


47 1 


8 


799892 


55 


8 


804003 


55 


8 


808076 


54 


8 


812111 


54 i 


9 


799961 


62 


9 


804071 


.62 


9 


808143 


61 


9 


812178 


60 1 


6310 


800029 




G370 


804139 




6480 


808211 




6490 


812245 


i 


1 


800098 


7 


1 


804208 


7 


1 j 808279 


7 


1 


812312 


7 


2 


800167 


14 


2 


804276 


14 


2 


808346 


14 


2 


812378 


13 i 


3 


800236 


21 


3 


804344 


21 


3 


808414 


20 


3 


812445 


20 i 


4 


800305 


28 


4 


804412 


27 


4 


808481 


27 


4 


812512 


27 1 


5 


800373 


34 


5 


804480 


34 


5 


808549 


34 


5 1 812579 


33 ! 


6 


800442 


41 


6 


804548 


41 


6 


808616 


41 


6 


812646 


40 ^ 


7 


800511 


48 


7 


804616 


48 


7 


808684 


48 


7 


812713 


47 i 


8 


800580 


55 


8 


804685 


55 


8 


808751 


54 


8 


812780 


54 


9 


800648 


62 


9 


804753 


62 


9 


808818 


61 


9 


812847 


60 


6320 


800717 




6380 


804821 




6440 


808886 




6500 


812913 




1 


800786 


7 


1 


804889 


7 


1 


808953 


7 


1 


812980 


7 


2 


800854 


14 


2 


804957 


14 


2 


809021 


13 


2 


813047 


13 


3 


800923 


21 


3 


805025 


20 


3 


809088 


20 


3 


813114 


20 


4 


800992 


28 


4 


805093 


27 


4 


809156 


27 


4 


813181 


27 


5 


801060 


34 


5 


805161 


34 


5 


809223 


34 


5 


813247 


33 


6 


801129 


41 


6 


805229 


41 


6 


809290 


40 


6 


813314 


40 


7 


801198 


48 


7 


805297 


48 


7 


809358 


47 


7 


813381 


47 


8 


801266 


55 


8 


805365 


54 


8 


809425 


54. 


8 


813448 


54 


9 


801335 


62 


9 


805433 


61 


9 


809492 


61 


9 


813514 


60 


6330 


801404 




6390 


805501 




6450 


809560 




6510 


813581 




1 


1 801472 


7 


1 


805569 


7 


1 


809627 


7 


1 


813648 


7 


2 


1 801541 


14 


2 


805637 


14 


2 


809694 


13 


2 


813714 


13 


3 


! 801609 


21 


3 


805705 


20 


3 


809762 


20 


3 


813781 


20 


4 


1801678 


27 


4 


805773 


27 


4 


809829 


27 


4 


813848 


27 


5 


! 801747 


34 


5 


805841 


34 


5 


809896 


34 


5 


813914 


33 


6 


801815 


41 


6 


805908 


41 


6 


809964 


40 


6 


813981 


40 


7 


801884 


48 


7 


805976 


48 


7 


810031 


47 


7 


814048 


47 


8 


801952 


55 


8 


806044 


54 


8 


810098 


54 


8 


814114 


54 


9 


1 802021 


62 


9 


806112 1 61 


1 ^ 


810165 


61 


9 


814181 


60 



26 



LOGARITHMS OF NUMBERS. 



No. 


l-og. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Pare. 


No. 1 Log. 


Prop. 
Part. 


No. 1 Log. 


Prop. 
Pari. 


6620 


814248 




6580 


818226 




6640 


822168 




1 

6700 826075 




1 


814314 


7 


1 


818292 


7 


1 


822233 


7 


1 


826140 


6 


2 


814381 


13 


2 


818358 


13 


2 


822299 


13 


2 


826204 


13 


3 


814447 


20 


3 


818424 


20 


3 


822364 


20 


3 


826269 


19 


4 


814614 


26 


4 


818490 


26 


4 


822430 


26 


4 


826334 


26 


5 


814581 


33 


5 


818666 


33 


5 


822496 


33 


5 


826399 


32 


6 


814647 


40 


6 


818622 


40 


6 


822560 


39 


6 


826464 


39 


• 7 


814714 


46 


7 


818688 


46 


7 


822626 


46 


7 


826528 


45 


8 


814780 


53 


8 


818764 


63 


8 


822691 


62 


8 


826593 


52 


! y 


814847 


60 


9 


818819 


59 


9 


822756 


69 


9 


826658 


58 


1 6530 


814913 




6590 


818885 




6660 


822823 




6710 


826722 




1 


814980 


7 


1 


818951 


7 


1 


822887 


7 


1 


826787 6 


2 


816046 


13 


2 


819017 


13 


2 


822952 


13 


2 


8268521 13 


3 


816113 


20 


3 


819083 


20 


3 


823018 


20 


3 


8269171 19 


! 4 


815179 


26 


4 


819149 


26 


4 


823083 


26 


4 


826981 ! 26 


5 


816246 


33 


5 


819215 


33 


5 


823148 


33 


5 


827046 32 


6 


816312 


40 


6 


819281 


40 


6 


823213 


39 


6 


827111: 39 


7 


816378 


46 


7 


819346 


46 


7 


823279 


46 


7 


827175: 45 


8 


816446 


63 


8 


819412 


53 


8 


823344 


52 


8 


827240: 52 


9 


815511 


60 


9 


819478 


59 


9 


823409 


59 


9 


827306 1 58 


6540 


816678 




6600 


819644 




6660 


823474 i 


6720 


827369 i 


1 


816644 


7 


1 


819610 


7 


1 


823539 


7 


1 


827434 i 6 


2 


815711 


13 


>) 


819676 


13 


2 


823605 


13 


2 


827498! 13 


3 


815777 


20 


3 


819741 


20 


3 i 823670 


20 


3 


827563] 19 


4 


816843 


26 


4 


819807 


26 


4 823735 1 26 


4 


827628 j 26 


5 


815910 


33 


5 


819873 


33 


5 18238001 33 


5 


827692! 32 


6 


816976 


40 


6 


819939 


40 


6 


823865 


39 


6 


827757; 39 


7 


816042 


46 


7 


820004 


46 


7 


823930 


46 


7 


827821 j 45 


8 


816109 


53 


8 


820070 


63 


8 


823996 


52 


8 


827886' ^2 


9 


816176 


60 


9 


820136 


59 


>9 


824061 


59 


9 


827951 j 58 


6550 


816241 




6610 


820201 




6670 


824126 




6730 


828015 1 


1 


816308 


7 


1 


820267 


7 


1 


824191 


6 


1 


828080 i 6 


2 


816374 


13 


2 


820333 


13 


2 


824256 


13 


2 


828144 


13 


3 


816440 


20 


3 


820399 


20 


3 


824321 


19 


3 


828209 


19 


4 


816506 


26 


4- 


820464 


26 


4 


824386 


26 


4 


828273 


26 


5 


816573 


33 


5 


820630 


33 


5 


824461 


32 


5 


828338 


32 


6 


816639 


40 


6 


820695 


40 


6 


824516 


39 


6 


828402 


39 


7 


816705 


46 


7 


820661 


46 


7 


824681 


45 


7 


828467 


45 


8 


816771 


63 


8 


820727 


53 


'8 


824646 


52 


8 


828531 


52 


9 


816838 


60 


9 


820792 


59 


9 


824711 


58 


9 


828595 


58 


6660 


816904 




6620 


820858 




16680 


824776 




6740 


828660 




1 


816970 


7 


1 


820924 


7 


1 


824841 


6 


1 


828724 


6 


2 


817036 


13 


2 


820989 


13 


2 


824906 


13 


2 


828789 


13 


3 


817102 


20 


3 


821055 


20 


3 


824971 


19 


3 


828853 


19 


4 


817169 


26 


4 


821120 


26 


4 


825036 


26 


4 


828918 


26 


5 


817235 


33 


5 


821186 


33 


5 


825101 


32 


5 


828982 


32 


6 


817301 


40 


6 


821251 


40 


6 


825166 


39 


6 


829046 


39 


1 7 


817367 


46 


7 


821317 


46 


7 


825231 


46 


7 


829111 


46 


8 


817433 


53 


8 


821382 


53 


8 


825296 


52 


8 


829175 


52 


9 


817499 


59 


9 


821448 


59 


! 9 


826361 


68 


9 


829239 


58 


6570 


817566 




6630 


821614 




6690 


825426 




6750 


829304 




1 


817631 


7 


1 


821579 


7 


1 


826491 


6 


1 


829368 i 6 


2 


817698 


13 


2 ' 821644 


13 


2 


826556 


13 


2 


829432} 13 


3 


817764 


20 


3 821710 


20 


3 


825621 


19 


3 


829497 1 19 


4 


817830 


26 


4 821775 1 26 


4 


825686 


26 


4 


8295611 26 


5 


817896 


33 


5 821841 33 


6 


825751 


32 


6 


829625 1 32 


6 


817962 


40 


6 821906; 39 


6 


825815 


39 


6 


829690 39 


7 


818028 


46 


7 821972 i 46 


7 


825880 


45 


7 


829754 45 


8 


818094 


53 


8 822037 62 


! 8 


825945 


52 


8 


8298181 52 i 


9 


818160 


59 


9 822103 1 59 


9 ' 826010 


68 


9 


829882 1 58 | 



LOGARITHMS OF NUMBERS. 



27 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


u'.eo 


829947 




6820 


833784 




6880 


837588 




6940 


841359 




1 


830011 


6 


1 


833848 


6 


1 


837652 


6 


1 


841422 


6 


2 


830075 


13 


2 


833912 


13 


2 


837715 


13 


2 


841485 


13 


3 


830139 


19 


3 


833975 


19 


3 


837778 


19 


3 


841547 


19 


1 


830204 


26 


4 


834039 


26 


4 


837841 


25 


4 


841610 


25 


5 


830268 


32 


5 


834103 


32 


5 


837904 


32 


6 


841672 


31 


6 


830332 


38 


6 


834166 


38 


6 


837967 


38 


6 


841735 


38 


7 


830396 


45 


7 


834230 


45 


7 


838030 


44 


7 


841797 


44 


8 


830460 


51 


8 


834293 


51 


8 838093 


50 


8 


841860 


50 


9 


830525 


58 


9 


834357 


58 


9 


838156 


57 


9 


841922 


56 


6770 


830589 




6830 


834421 




6890 


838219 




6950 


841985 




1 


830653 


6 


1 


834484 


6 


1 


838282 


6 


1 


842047 


6 


2 


830717 


13 


2 


834548 


13 


2 


838345 


13 


2 


84211U 


12 


3 


830781 


19 


3 


834611 


19 


3 


838408 


19 


3 


842172 


19 


4 


830845 


26 


4 


834675 


26 


4 


838471 


25 


4 


842235 


25 


5 


830909 


32 


5 


834739 


32 


5 


838534 


32 


5 


842297 


31 


6 


830973 


38 


6 


834802 


38 


6 


838597 


38 


6 


842360 


37 


7 


831037 


45 


7 


834866 


45 


7 


838660 


44 


7 


842422 


44 


8 


831102 


51 


8 


834929 


51 


8 


838723 


50 


8 


842484 


50 


9 


831166 


58 


9 


834993 


58 


9 


838786 


57 


9 


842547 


56 


6780 


831230 




6840 


835056 




6900 


838849 




6960 


842609 




1 


831294 


6 


1 


835120 


6 


1 


838912 


6 


1 


842672 


6 


2 


831358 


13 


2 


835183 


13 


2 


838975 


13 


2 


842734 


12 


3 


831422 


19 


3 


835247 


19 


3 


839038 


19 


3 


842796 


19 


4 


831486 


26 


4 


835310 


26 


4 


839101 


25 ' 


4 


842859 


25 


5 831560 


32 


5 


835373 


32 


5 


839164 


31 j 


5 


842921 


31 


6 


831614 


38 


6 


835437 


38 


6 


839227 


38 


6 


842983 


37 


7 


831678 


45 


7 


835500 


45 


7 


839289 


44 


7 


843046 


44 


8 


831742 


51 


8 


835564 


51 


8 


839352 


50 


8 


843108 


50 


9 831806 


58 


9 


835627 


58 


9 


839415 


57 


9 


843170 


56 I 


6790 831870 




6850 


835691 




6910 


839478 




6970 


843233 




1 831934 


6 


1 


835754 


6 


1 


839541 


6 


1 


843295 


6 


2 i 831998 


13 


2 


835817 


13 


2 


839604 


13 


2 


843357 


12 


3 1 832062 


19 


3 


835881 


19 


3 


839667 


19 


3 


843420 


19 


4 


832126 


26 


4 


835944 


26 


. 4 


839729 


25 


4 


843482 


25 


5 


832189 


32 


5 


836007 


32 


5 


839792 


31 


5 


843544 


31 


6 


832253 


38 


6 


836071 


38 


6 


839855 


38 


6 


843606 


37 


7 


832317 


45 


7 


836134 


45 


7 


839918 


44 


7 


843669 


43 


8 


832381 


51 


8 


836197 


51 


8 


839981 


50 


8 


843731 


50 ! 


9 


832445 


58 


9 


836261 


58 


9 


840043 


57 


9 


843793 


56 1 


6800 


832509 




6860 


836324 




6920 


840106 




6980 


843855 




1 


832573 


6 


1 


836387 


6 


1 


840169 


6 


1 


843918 


6 


2 


832637 


13 


2 


836451 


13 


2 


840232 


13 


2 


843980 


12 


3 


832700 


19 


3 


836514 


19 


3 


840294 


19 


3 


844042 


19 


4 


832764 


26 


4 


836577 


26 


4 


840357 


25 


4 


844104 


25 


5 


832828 


32 


5 


836641 


32 


5 


840420 


31 


5 


844166 


31 


6 ! 832892 


38 


6 


836704 


38 


6 


840482 


38 


6 


844229 


37 


7 


832956 


45 


7 


836767 


45 


7 


840545 


44 


7 


844291 


43 


8 


833020 


51 


8 


836830 


51 


8 


840608 


50 


8 


844353 


50 


9 


833083 


58 


9 


836894 


58 


9 


840671 


57 


9 


844415 


56 


6810 


833147 




6870 


836957 




6930 


840733 




6990 


844477 




1 


833211 


6 


1 


837020 


6 


1 


840796 


6 


1 


844539 


6 


2 


833275 


13 


2 


837083 


13 


2 


840859 


13 


2 


844601 


12 


3 


833338 


19 


3 


837146 


19 


3 


840921 


19 


3 


844664 


19 


4 


833402 


26 


4 


837210 


25 


4 


840984 


25 


4 


844726 


25 


5 


833466 


32 


5 


837273 


32 


5 


841046 


31 


5 


844788 


31 


6 


833530 


38 


6 


837336 


38 


6 


841109 


38 


6 


844850 


37 


7 


833593 


45 


7 


837399 


44 


7 


841172 


44 


7 


844912 


43 


8 i 833657 


51 


8 


837462 


51 


8 


841234 


50 


8 


844974 


50 


9 1 838721 


58 


9 


837525 


57 


9 


841297 


56 


9 ' 845036 


56 



28 



LOGARITHMS OF NUMBERS. 



NO, 


Log. 


Prop. 
Part. 


No. ! 


Log. 1 


Prop. 
Part. ; 


No. 


.Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


7000 


845098 




70601 


848805 




7120 


852480 




7180 


866124 




1 


845160 


6 


1 


848866 


6 


1 


862641 


6 


1 


856185 


6 


2 


845222 


12 


21 


848928 


12 


2 


852602 


12 


2 


866246 


12 


3 


845284 


19 


3; 


848989 


18 


3 


852663 


18 


3 


856306 


18 


4 


845346 


25 


4j 


849051 


25 


4 


852724 


24 


4 


856366 


24 


5 


845408 


31 


6 


849112 


31 


5 


852785 


30 


5 


856427 


30 


6 


845470 


37 


6 


849174 


37 


6 


852846 


87 


6 


856487 


36 


7 


845532 


43 


7 


849235 


43 


7 


852907 


48 


7 


856548 


42 


8 


845594 


50 


8 


849296! 49 ' 


8 


852968 


49 


8 


856608 


48 


9 


845656 


66 


9 


849368 


66 


9 


858029 


55 


9 


856668 


54 


7010 


845718 




7070 


849419 




7130 


853090 


1 


7190 


856729 




1 


845780 


6 


1 


849481 


6 


1 


853150 


6 


1 


856789 


6 


2 


845842 


12 


2 


849542 


12 1 


2 


853211 


12 


2 


856850 


12 


3 


845904 


19 


8 


849604 


18 1 


3 


853272 


18 1 


3 


856910 


18 


4 


845966 


25 


4 


849665 


25 1 


4 


853333 


24 


4 


856970 


24 


5 


846028 


31 


5 


849726 


31 j 


5 


853394 


80 


5 


857031 


30 


6 


846090 


37 


6 


849788 


37 1 


6 


863455 


37 


6 


867091 


30 


7 


846151 


43 


7 


849849 


43 i 


7 


853516 


43 


7 


857161 


42 


8 


846213 


50 


8 


849911 


49 


8 


853576 


49 


8 


857212 


48 


9 


846275 


56 


9 


849972 


55 


9 


853637 


56 


9 


857272 


64 


7020 


846337 




7080 


850033 




7140 


853698 




7200 


867332 




1 


846399 


6 


1 


850096 


6 


1 


853759 


6 


1 


867393 


6 


2 


846461 


12 


2 


850156 1 12 ! 


2 


853820 


12 


2 


857453 


12 


3 


84G523 


19 


3 


850217 18 ■ 


3 


853881 


18 


1 3 


857513 


18 


4 


846584 


25 


4 


850279 


25 ' 


4 


868941 


24 


4 


857574 


24 


5 


846646 


31 


6 


850340 


31 


6 


854002 


30 


5 


867634 


80 


6 


846708 


37 


6 


850401 


37 


6 


854063 


87 


6 


857694 


86 


7 


846770 


43 


7 


850462 


43 


7 


854124 


48 


7 


857764 


42 


8 


846832 


50 


8 


850524 


49 


8 


864185 


49 


8 


857815 


48 1 


9 


846894 


56 


9 


850585 


65 


9 


854245 


55 


9 


867875 


64 


7030 


846955 




7090 


850646 




7160 


854306 


' 


7210 


857935 


1 


1 


847017 


6 


1 


860707 


6 


1 


854867 


6 


1 


857995 


6 i 


2 


847079 


12 


2 


850769 


12 


2 


854427 


12 


2 


858066 


12 


3 


847141 


19 


3 


850830 


18 


3 


854488 


18 


8 


858116 


18 


4 


847202 


25 


4 


850891 


25 


4 


854649 


24 


4 


858176 


24 


5 


847264 


31 


6 


850952 


31' 


5 


854610 


30 


5 


858236 


80 


6 


847326 


37 


6 


851014 


37 


6 


854670 


36 


6 


858297 


36 


7 


847388 


43 


— 


851075 


43 


7 


854781 


42 


7 


858367 


42 


8 


847449 


60 


8 


861136 


49 


8 


854792 


48 


8 


858417 


48 


9 


847511 


56 


9 


851197 


55 


9 


854862 


64 


9 


858477 


64 


7040 


847573 




7100 


851268 




7160 


864918 




7220 


868587 




1 


847634 


6 


1 


851320 


6 


1 


864974 


6 


1 


858697 


6 


2 


847696 


12 


2 


851381 


12 


2 


856034 


12 


2 


858657 


12 


3 


847758 


18 


3 


851442 


18 


3 


855096 


18 


3 


868718 


18 


4 


847819 


25 


4 


861503 


26 


4 


855166 


24 


4 


858778 


24 


5 


847881 


31 


5 


851664 


31 


5 


856216 


30 


6 


868888 


80 


6 


847943 


37 


6 


851625 


37 


6 


865277 


36 


6 


858898 


86 


7 


848004 


43 


7 


851686 


43 


7 


855887 


42 


7 


858958 


42 


8 


848066 


49 


8 


851747 


49 


8 


855398 


48 


8 


859018 


48 


9 


848127 


65 


9 


851808 


55 


9 


866469 


54 


9 


859078 


64 


7050 


848189 




7110 


861870 




7170 


856619 




7280 


859138 




1 


848251 


6 


1 


861931 


6 


1 


856680 


6 


1 


859198 


6 


2 


848312 


12 


2 


861992 


12 


2 


856640 


12 


2 


859258 


12 


3 


848374 


18 


3 


862063 


18 


3 


856701 


18 


8 


859318 


18 


4 


848435 


25 


4 


862114 


26 


4 


865761 


24 


4 


859378 


24 


5 


848497 


31 


5 


862175 


31 


6 


855822 


80 


5 


859488 


30 


6 


848559 


37 


6 


862236 


37 


6 


865882 


36 


6 


859499 


36 


7 


848620 


43 


7 


862297 


43 


7 


855943 


42 


7 


859559 


42 


8 


848682 


49 


8 


852368 


49 


8 


856003 


48 


8 


859619 


48 


9 


848743 


55 


9 


862419 


65 


9 


866064 


54 


9 


859679 


54 



LOGARITHMS OF NUMBERS. 



29 



No. 


Log. 


Prop. 
Part. 


NO. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


7240 


859739 




7300 


863323 


1 


7360 


866878 




7420 


870404 




1 


859799 


6 


1 


863382 


6 1 


1 


866937 


6 


1 


870462 


6 


1 2 


859858 


12 


2 


863442 


12 


2 


866996 


12 


2 


870521 


12 


3 


859918 


18 


3 


863501 


18 


3 


867055 


18 


3 


870579 


18 


4 


859978 


24 


4 


863561 


24 


4 


867114 


24 


4 


870638 


24 


5 


860038 


30 


5 


863620 


30 


5 


867173 


29 


5 


870696 


29 


6 


860098 


36 


6 


863680 


36 


6 


867232 


35 


6 


870755 


35 


7 


860158 


42 


7 


863739 


42 


7 


867291 


41 


7 


870813 


41 


8 


860218 


48 


8 


863798 


48 


8 


867350 


47 


8 


870872 


47 


9 


860278 


54 


9 


863858 


54 ! 


9 


867409 


53 


9 


870930 


53 


7250 


860338 




7310 


863917 




7370 


867467 




7430 


870989 




1 


860398 


6 


1 


863977 


6 ! 


1 


867526 


6 


1 


871047 


6 


2 


860458 


12 


2 


864036 


12 


2 


867585 


12 


2 


871106 


12 


3 


860518 


18 


3 


864096 


18 


3 


867644 


18 


3 


871164 


18 


4 


860578 


24 


4 


864155 


24 


4 


867703 


24 


4 


871223 


24 


5 860637 


30 


5 


864214 


30 


5 


867762 


29 


5 


871281 


29 


6 ! 860697 


36 


6 


864274 


36 


6 


867821 


35 


6 


871339 


.35 


7 


860757 


42 


7 


864333 


42 


7 


867880 


41 


7 


871398 


41 


8 


860817 


48 


8 


864392 


48 


8 


867939 


47 


8 


871456 


47 


9 


860877 


54 


9 


864452 


54 


9 


867998 


53 


9 


871515 


53 


7260 


860937 




7320 


864511 




7380 


868056 




7440 


871573 




1 


860996 


6 


1 


864570 


6 


1 


868115 


6 


1 


871631 


6 


2 


861056 


12 


2 


864630 


12 


2 


868174 


12 


2 


871690 


12 


3 1861116 


18 


3 


864689 


18 


3 


868233 


18 


3 


871748 


18 


4 1 861176 


24 


4 


864748 


24 


4 


868292 


24 


4 


871806 


23 


5 861236 


30 


5 


864808 


30 


5 


868350 


29 


5 


871865 


29 


6 861295 


36 


6 


864867 


36 


6 


868409 


35 


6 


871923 


35 


7 861355 


42 


7 


864926 


42 


7 


868468 


41 


7 


871981 


41 


8 861415 


48 


8 


864985 


48 


8 


868527 


47 


8 


872040 


47 


9 j 861475 


54 


9 


865045 


54 


9 


868586 


53 


9 


872098 


53 


7270 


861534 




7330 


865104 




7390 


868644 




17450 


872156 




1 


861594 


6 


1 


865163 


6 


1 


868703 


6 


1 


872215 


6 


2 


861654 


12 


2 


865222 


12 


2 


868762 


12 


2 


872273 


12 


3 


861714 


18 


3 


865282 


18 


3 


868821 


18 


3 


872331 


18 


4 


861773 


24 


4 


865341 


24 


4 


868879 


24 


4 


872389 


23 


5 


861833 


30 


5 


865400 


30 


5 


868938 


29 


5 


872448 


29 


6 


861893 


36 


6 


865459 


36 


6 


868997 


35 


6 


872506 


35 


7 


861952 


42 


7 


865518 


42 


7 


869056 


41 


7 


872564 


41 


8 


862012 


48 


8 


865578 


48 


8 


869114 


47 


8 


872622 


47 


9 


862072 


54 


9 


865637 


54 


9 


869173 


53 


9 


872681 


53 


7280 


862131 




7340 


865696 




7400 


869232 




7460 


872739 




1 


862191 


6 


1 


865755 


6 


1 


869290 


6 


1 


872797 


6 


2 


862251 


12 


2 


865814 


12 


2 


869349 


12 


2 


872855 


12 


3 


862310 


18 


3 


865874 


18 


3 


869408 


18 


3 


872913 


18 


4 


862370 


24 


4 


865933 


24 


4 


869466 


24 


4 


872972 


23 


5 


862430 


30 


5 


865992 


30 


5 


869525 


29 


5 


873030 


29 


6 


862489 


36 


6 


866051 


36 


6 


869584 


35 


6 


873088 


35 1 


7 


862549 


42 


7 


866110 


42 


7 


869642 


41 


7 


873146 


41 


8 


862608 


48 


8 


866169 


48 


8 


869701 


47 


8 


873204 


47 


9 


862668 


54 


9 


866228 


54 


9 


869760 


53 


9 


873262 


53 


7290 


862728 




7350 


866287 




7410 


869818 




7470 


873321 




1 


862787 


6 


1 


866346 


6 


1 


869877 


6 ' 


1 


873379 


6 


2 


862847 


12 


2 


866405 


12 


2 


869935 


12 


2 


873437 


12 


3 


862906 


18 


3 


866465 


18 


3 


869994 


18 


3 


873495 


18 


4 


862966 


24 


4 


866524 


24 


4 


870053 


24 j 


4 


873553 


23 


5 


863025 


30 


5 


866583 


30 


6 


870111 


29 


5 


873611 


29 


6 


863085 


36 


6 


866642 


35 


6 


870170 


35 


6 


873669 


35 


7 


863144 


42 


7 


866701 


41 


7 


870228 


41 


7 


873727 


41 


8 


863204 


48 


8 


866760 


47 


8 


870287 


47 


8 


873785 


47 


9 


863263 


54 


9 


866819 


63 


9 


870345 


53 


9 


873844 


53 



30 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. Log. f-^| 


7480 


873902 




7540 


877371 




7600 


880814 




7660 884229 




1 


873960 


6 


1 


877429 


6 


1 


880871 


6 


1 ' 884285 


6 


2 


874018 


12 


2 


877486 


12 


2 


880928 


11 


2 884342 


11 


3 


874076 


17 


3 


877544 


17 


3 


880985 


17 


3 884399 


17 


4 


874134 


23 


4 


877602 


23 


4 


881042 


23 


4 884455 


23 


5 


874192 


29 


5 


877659 


29 


5 


881099 


28 


5 884512 


28 


6 


874250 


35 


6 


877717 


34 


6 


881156 


34 


6 ' 884569 


34 


7 


874308 


41 


7 


877774 


40 


7 


881213 


40 


7 884625 


40 


8 


874366 


46 


8 


877832 


46 


8 


881270 


46 


8 884682 


46 


9 


874424 


52 


9 


877889 


52 


9 


881328 


51 


9 ; 884739 


51 


7490 


874482 




7550 


877947 




7610 


881385 




7670 ! 884795 




1 


874540 


" 6 


1 


878004 


6 


1 


881442 


6 


1 1 884852 


6 


2 


874598 


12 


2 


878062 


12 


2 


881499 


11 


2 1 884909 


11 


3 


874656 1 17 


3 


878119 


17 


3 


881556 


17 


3 1 884965 


17 


4 


874714 


23 


4 


878177 


23 


4 


881613 


23 


4 1 885022 


23 


5 


874772 


29 


5 


878234 


29 


5 


881670 


28 


5 i 885078 


28 


6 


874830 


35 


6 


878292 


34 


6 


881727 


34 


G j 885135 


34 


7 


874887 


41 


7 


878349 


40 


7 


881784 


40 


7 : 885192 


40 


8 


874945 


46 


8 


878407 


46 


8 


881841 


46 


_ 8 ■ 885248 


46 


9 


875003 


52 


9 


878464 


52 


9 


881898 


51 


9 j 885305 


51 


7500 


875061 




7560 


878522 




7620 


881955 




7680 885361 




1 


875119 


6 


1 


878579 


6 


1 


882012 


6 


1 ' 885418 


6 


2 


875177 


12 


2 


878637 


12 


2 


882069 


11 


2 ' 885474 


11 


3 


875235 


17 


3 


878694 


17 


3 


882126 


17 


3 885531 


17 


4 


875293 


23 


4 


878751 


23 


C: 882183 


23 


4 885587 


23 


5 


875351 


29 


5 


878809 


29 


5 ! 882240 


28 


5 885644 


28 


6 


875409 


35 


6 


878866 


34 


6 


882297 


34 


6 885700 


34 


7 


875466 


41 


7 


878924 


40 


7 


882354 


40 


7 885757 


39 


8 


875524 


46 


8 


878981 


46 


8 


882411 


46 


8 885813 


45 


9 


875582 


52 


9 


879038 


52 


9 


882468 


51 


9 885870 


51 


7510 


875640 




7570 


879096 




7630 


882524 




7690 , 885926 




1 


875698 


6 


1 


879153 


6 


1 


882581 


6 


1 ! 885983 


6 


2 


875756 


12 


2 


879211 


12 


2 


882638 


11 


2 


886039 


11 


3 


875813 


17 


3 


879268 


17 


3 


882695 


17 


3 


886096 


17 


4 


875871 


23 


4 


879325 


23 


4 


882752 


23 


4 


886152 


23 


5 


875929 


29 


5 


879383 


29 


5 


882809 


28 


5 


886209 


28 


6 


875987 


35 


6 


879440 


34 


6 


882866 


34 


6 


886265 


34 


7 


876045 


41 


7 


879497 


40 


7 


882923 


40 


7 


886321 


39 


8 


876102 


46 


8 


879555 


46 


8 


882980 


46 


8 


886378 


45 


9 


876160 


52 


9 


879612 


52 


9 


883037 


51 


9 


886434 


51 


7520 


876218 




7580 


879669 




7640 


883093 




7700 


886491 




1 


876276 


6 


1 


879726 


6 


1 


883150 


6 


1 


886547 


6 


2 


876333 


12 


2 


879784 


11 


2 


883207 


11 


2 


886604 


11 


3 


876391 


17 


3 


879841 


17 


3 


883264 


17 


3 


886660 


17 


4 


876449 


23 


4 


879898 


23 


4 


883321 


23 


4 


886716 


23 


5 


876507 


29 


5 


879956 


28 


5 


883377 


28 


5 


886773 


28 


.6 


876564 


34 


6 


880013- 


34 


6 


883434 


34 


6 


886829 


34 


7 


876622 


40 


7 


880070 


40 


7 


883491 


40 


7 


886885 


39 


8 


876680 


46 


8 


880127 


46 


8 


883548 


46 


8 


886942 


45 


9 


876737 


52 


9 


880185 


51 


9 


883605 


51 


9 


886998 


51 


7530 


876795 




7590 


880242 




7650 


883661 




7710 


887054 




1 


876853 


6 


1 


880299 


6 


1 


883718 


6 


1 


887111 


6 


2 


876910 


12 


2 


880356 


11 


2 


883775 


11 


2 


887167 


11 


3 


876968 


17 


3 


880413 


17 


3 


883832 


17 


3 


887223 


17 


4 


877026 


23 


4 


880471 


23 


4 


883888 


23 


4 


887280 


23 


5 


877083 


29 


5 ; 880528 


28 


5 


883945 


28 


5 


887336 


28 


6 


877141 


34 


6 ' 880585 


34 


6 


884002 


34 


6 


887392 


34 


, 7 


877198 


40 


7 880642 


40 


7 


884059 


40 


7 


887449 


39 


8 


877256 


46 


8 880699 


46 


8 


884115 


46 


8 


887505 


45 


9 1 877314 


52 


9 880756 


51 


9 


884172 


61 


9 887561 


51 



LOGARITHMS OF NUMBERS. 



31 



No. 


Log. 


Prop. 
Part. 


No. 


^0^- !p^a?r 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


7720 


887617 




7780 


890980 




7840 


894316 




7900 


897627 




1 


887674 


6 


1 


891035 


6 


1 


894371 


6 


1 


897682 


6 


2 


887730 


11 


2 


891091 


11 


2 


894427 


11 


2 


897737 


11 i 


3 


887786 


17 


3 


891147 


17 


3 


894482 


17 


3 


897792 


17 i 


4 


887842 


23 


4 


891203 


22 


4 


894538 


22 


4 


897847 


22 


5 


887898 


28 


5 


891259 


28 


5 


894593 


27 


5 


897902 


27 


6 


887955 


34 


6 


891314 


34 


6 


894648 


33 


6 


897957 


33 


7 


888011 


39 


7 


891370 


39 


7 


894704 


39 


7 


898012 


39 


8 


888067 


45 


8 


891426 


45 


8 


894759 


44 


8 


898067 


44 


9 


888123 


51 


9 


891482 


50 


9 


894814 


50 


9 


898122 


50 


7730 


888179 




7790 


891537 




7850 


894870 




7910 


898176 




1 


888236 





1 


891593 


6 


1 


894925 


6 


1 


898231 


6 


2 


888292 


11 


2 


891649 


11 


2 


894980 


11 


2 


898286 


11 


3 


888348 


17 


3 


891705 


17 


3 


895036 


17 


3 


898341 


17 


4 


888404 


22 


4 


891760 


22 


4 


895091 


22 


4 


898396 


22 


5 


888460 


28 


5 


891816 


28 


5 


895146 


27 


5 


898451 


27 


6 


888516 


34 


6 


891872 


33 


6 


895201 


33 


6 


898506 


33 


7 


888573 


39 


7 


891928 


39 


7 


895257 


39 


7 


898561 


39 


8 


888629 


45 


8 


891983 


44 


8 


895312 


44 


8 


898615 


44 


9 


888685 


50 


9 


892039 


50 


9 


895367 


50 


9 


898670 


50 


7740 


888741 




7800 


892095 




7860 


895423 




7920 


898725 




1 


888797 


6 


1 


892150 


6 


1 


895478 


6 


1 


898780 


5 


2 


888853 


11 


2 


892206 


11 


2 


895533 


11 


2 


898835 


11 


3 


888909 


17 


3 


892262 


17 


3 


895588 


17 


3 


898890 


17 


4 


888965 


22 


4 


892317 


22 


4 


895643 


22 


4 


898944 


22 


5 


889021 


28 


5 


892373 


28 


5 


895699 


27 


5 


898999 


27 


6 


889077 


34 


6 


892429 


33 


6 


895754 


33 


6 


899054 


33 


7 


889134 


39 


7 


892484 


39 


7 


895809 


39 


7 


899109 


38 


8 


889190 


45 


8 


892540 


44 


8 


895864 


44 


8 


899164 


44 


9 


889246 


50 


9 


892595 


50 


9 


895920 


50 


9 


899218 


50 


7750 


889302 




7810 


892651 




7870 


895975 




7930 


899273 




1 


889358 


6 


1 


892707 


6 


1 


896030 


6 


1 


899328 


5 


2 


889414 


11 


2 


892762 


11 


2 


896085 


11 


2 


899383 


11 


3 


889470 


17 


3 


892818 


17 


3 


896140 


17 


3 


899437 


17 


4 


889526 


22 


4 


892873 


22 


4 


896195 


22 


4 


899492 


22 


5 


889582 


28 


5 


892929 


28 


5 


896251 


27 


5 


899547 


27 


6 


889638 


34 


6 


892985 


33 


6 


896306 


33 


6 


899602 


33 


7 


889694 


39 


7 


893040 


39 


7 


896361 


39 


7 


899656 


38 


8 


889750 


45 


8 


893096 


44 


8 


896416 


44 


8 


899711 


44 


9 


889806 


50 


9 


893151 


50 


9 


896471 


50 


9 


899766 


50 


7760 


889862 




7820 


893207 




7880 


896526 




7940 


899820 




1 


889918 


6 


1 


893262 


6 


1 


896581 


6 


1 


899875 


5 


2 


889974 


11 


2 


893318 


11 


2 


896636 


11 


2 


899930 


11 


3 


890030 


17 


3 


893373 


17 


3 


896692 


17 


3 


899985 


17 


4 


890086 


22 


4 


893429 


22 


4 


896747 


22 


4 


900039 


22 


6 


890141 


28 


5 


893484 


28 


5 


896802 


27 


5 j 900094 j 27 | 


6 


890197 


34 


6 


893540 


33 


6 


'896857 


33 


6 , 900149 1 33 \ 


7 


890253 


39 


7 


893595 


39 


7 


896912 


39 


7 1 900203 38 j 


8 


890309 


45 


8 


893651 


44 


8 


896967 


44 


8 900258 1 44 | 


9 


890365 


50 


9 


893706 


50 


9 


897022 


50 


9 i 900312 


50 


7770 


890421 




7830 


893762 




7890 


897077 




7950 


900367 




1 


890477 


6 


1 


893817 


6 


1 


897132 


6 


1 


900422 


5 


2 ' 890533 


11 


2 


893873 


11 


2 


897187 


11 


2 


900476 


11 


3 j 890589 


17 


3 


893928 


17 


3 


897242 


17 


3 


900531 


17 


4 


890644 


22 


4 


893984 


22 


4 


897297 


22 


4 


900586 


22 


5 


890700 


28 


5 


894039 


28 


5 


897352 


27 


5 


900640 


27 


6 


890756 


34 


6 


894094 


33 


6 


897407 


33 


6 


900695 


33 1 


7 


890812 


39 


7 


894150 


39 


7 


897462 


39 


7 


900749 38 


8 


890868 


45 


8 


894205 


44 


8 


897517 


44 


8 


900804; 44 


9 


890924 


50 


9 


894261 


50 


9 897572 


50 


9 


900858, 50 



32 



LOGARITHMS OF NUMBERS. 



Nc. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


7960 


900913 




8020 


904174 




8080 


907411 




8140 


910624 




1 


900968 


5 


1 


904228 


5 


1 


907465 


5 


1 


910678 


5 


2 


901022 


11 


2 


904283 


11 


2 


907519 


11 


2 


910731 


11 


3 901077 


16 


3 


904337 


16 


3 


907573 


16 


3 


910784 


16 


4 


901131 


22 


4 


904391 


22 


4 


907626 


22 


4 


910838 


21 


5 


901186 


27 


5 


904445 


27 


5 


907680 


27 


5 


910891 


27 


6 


901240 


33 


6 


904499 


32 


6 


907734 


32 


6 


910944 


32 


7 


901295 


38 


7 


904553 


38 


7 907787 


38 


7 


910998 


37 


8 


901349 


44 


8 


904607 


43 


8 


907841 


43 


8 


911051 


43 


9 


901404 


49 


9 


904661 


49 


9 


907895 


49 


9 


911104 


48 


7970 


901458 




8030 


904715 




8090 


907948 




8150 


911158 




1 


901513 


5 


1 


904770 


5 


1 


908002 


5 


1 


911211 


6 


2 


901567 


11 


2 


904824 


11 


2 


908056 


11 


2 


911264 


11 


3 


901622 


16 


3 


904878 


16 


3 


908109 


16 


3 


911317 


16 


4 


901676 


22 


4 


904932 


22 


4 


908163 


22 


4 


911371 


21 


5 


901731 


27 


5 


904986 


27 


5 


908217 


27 


5 


911424 


27 


6 


901785 


33 


6 


905040 


32 


6 


908z70 


32 


6 


911477 


32 


7 


901840 


38 


7 


905094 


38 


7 


908324 


38 


7 


911530 


37 


8 


901894 


44 


8 


905148 


43 


8 


908378 


43 


8 


911584 


42 


9 


901948 


49 


9 


905202 


49 


9 


908431 


49 


9 


911637 


48 


7980 


902003 




8040 


905256 




8100 


908485 




8160 


911690 




1 


902057 


5 


1 


905310 


5 


1 


908539 


5 


1 


911743 


5 


2 


902112 


11 


2 


905364 


11 


2 


908592 


11 


2 


911797 


11 


3 


902166 


16 


3 


905418 


16 


3 


908646 


16 


3 


911850 


16 


4, 


902221 


22 


4 


905472 


22 


4 


908699 


21 


4 


911903 


21 


5 


902275 


27 


5 


905526 


27 


5 


908753 


27 


5 


911956 


27 


6 


902329 


33 


6 


905580 


32 


6 


908807 


32 


6 912009 


32 


7 


902384 


38 


7 


905634 


38 


7 


908860 


37 


7 


912063 


37 


8 


902438 


44 


8 


905688 


43 


8 


908914 


43 


8 


912116 


42 


9 


902492 


49 


9 


905742 


49 


9 


908967 


48 


9 


912169 


48 


7990 


902547 




8050 


905796 




8110 


909021 




8170 


912222 




1 ! 902601 


5 


1 


905850 


5 


1 


909074 


5 


1 


912275 


5 


2 ! 902655 


11 


2 


905904 


11 


2 


909128 


11 


2 


912328 


11 


3 i 902710 


16 


3 


005958 


16 


3 


909181 


16 


3 


912381 


16 


4 i 902764 


22 


4 


906012 


22 


4 


909235 


21 


4 


912435 


21 


5 1 902818 


27 


5 


906065 


27 


5 


909288 


27 


5 


912488 


27 


6 1 902873 


33 


6 


906119 


32 


6 


909342 


32 


6 


912541 


32 


7 902927 


38 


7 


906173 


38 


7 


909395 


37 


7 


912594 


37 


8 j 902981 


44 


8 


906227 


43 


8 


909449 


43 


8 


912647 


42 


9 


903036 


49 


i 9 


906281 


49 


9 


909502 


48 


9 


912700 


48 


8000 


903090 




8060 


906335 




8120 


909556 




8180 


912753 




1 


903144 


5 


1 


906389 


5 


1 


909609 


5 


1 


912806 


5 


2 


903198 


11 


2 


906443 


11 


2 


909663 


11 


2 


912859 


11 


3 


903253 


16 


3 


906497 


16 


3 


909716 


16 


3 


912913 


16 


4 


903307 


22 


4 


906550 


22 


4 


909770 


21 


4 


912966 


21 


5 


903361 


27 


5 


906604 


27 


5 


909823 


27 


5 


913019 


27 


6 


903416 


32 


6 


906658 


32 


6 


909877 


32 


6 


913072 


32 


7 


903470 


38 


7 


906712 


38 


7 


909930 


37 


7 


913125 


37 


8 


903524 


43 


8 


906766 


43 


8 


909984 


43 


8 


913178 


42 


9 


903578 


49 


9 


906820 


49 


9 


910037 


48 


9 


913231 


48 


8010 


903632 




8070 


906873 




8130 


910090 




8190 


913284 




1 


903687 


5 


1 


906927 


5 


1 


910144 


5 


1 


913337 


5 


2 


903741 


11 


2 


906981 


11 


2 


910197 


11 


2 


913390 


11 


3 


903795 


16 


3 


907035 


16 


3 


910251 


16 


3 


913443 


16 


4 


903849 


22 


4 


907089 


22 


4 


910304 


21 


4 


913496 


21 


1 5 


903903 


27 


5 


907142 


27 


5 


910358 


27 


6 


913549' 27 


i 6 


903958 


32 


6 


907196 


32 


6 


910411 


32 


6 


913602 1 32 


7 


904012 


38 


7 


907250 


38 


1 ^ 


910464 


37 


7 


913655 1 37 


8 


904066 


43 


8 


907304 


43 


i 8 


910518 


43 


8 


913708 42 


9 
1 


904120 


49 


9 


907358 


49 


9 


910571 


48 


9 


913761 48 



LOGAKITHMS OF NUMBERS. 



33 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. Log. 


Prop. 
Part. 


8200 


913814 




8260 


916980 




8320 


920123 




8380 923244 




1 


913867 


5 


1 


917033 


5 


1 


920175 


6 


1 i 923296 


5 


2 


913920 


11 


2 


917085 


11 


2 


920228 


10 


2 i 923348 


10 


3 


913973 


16 


3 


917138 


16 


3 


920280 


16 


3 ! 923399 


16 


4 


914026 


21 


4 


917190 


21 


4 


920332 


21 


4 ' 923451 


21 


5 


914079 


27 


5 


917243 


26 


5 


920384 


26 


5 923503 


26 


6 


914131 


32 


6 


917295 


31 


6 


920436 


31 


6 1 923555 


31 


7 


914184 


37 


7- 


917348 


37 


7 


920489 


36 


7 1 923607 


36 


8 


914237 


42 


8 


917400 


42 


8 


920541 


42 


8 1 923658 


42 


y 


914290 


48 


9 


917453 


47 


9 


920593 


47 


9 


923710 


47 


8210 


914343 




8270 


917505 




8330 


920645 




8390 


923762 




1 


914396 


5 


1 


917558 


5 


1 


920697 


5 


1 


923814 


5 


2 


914449 


11 


2 


917610 


11 


2 


920749 


10 


2 


923865 


10 


3 


914502 


16 


3 


917663 


16 


3 


920801 


16 


3 


923917 


16 


4 


914555 


21 


4 


917715 


21 


4 


920853 


21 


4 


923969 


21 


5 


914608 


27 


5 


917768 


26 


5 


920906 


26 


5 


924021 


26 


6 


914660 


32 


6 


917820 


31 


6 


920958 


31 


6 


924072 


31 


7 


914713 


37 


7 


917873 


37 


7 


921010 


36 


7 


924124 


36 


8 


914766 


42 


8 


917925 


42 


8 


921062 


42 


8 


924176 


42 


9 


914819 


48 


9 


917978 


47 


9 


921114 


47 


9 


924228 


47 


8220 


914872 




8280 


918030 




8340 


921166 




8400 


924279 




1 


914925 


5 


1 


91 8083 


5 


1 


921218 


5 


1 


924331 


5 


2 


914977 


11 


o 


918135 


11 


2 


921270 


10 


2 


924383 


10 


3 


915030 


16 


3 


918188 


16 


3 


921322 


16 


3 


924434 


15 


4 


915083 


21 


4 


918240 


21 


4 


921374 


21 


4 


924486 


21 


5 


915136 


27 


5 


918292 


26 


5 


921426 


26 


5 


^24538 


26 


6 


915189 


32 


6 


918345 


31 


6 


921478 


31 


6 


924589 


31 


7 


915241 


37 


7 


918397 


37 


7 


921530 


36 


7 


924641 


36 


8 


915294 


42 


8 


918450 


42 


8 


921582 


42 


8 


924693 


41 


9 


915347 


48 


9 


918502 


47 


9 


921634 


47 


9 


924744 


46 


8230 


915400 




8290 


918555 




8350 


921686 




8410 


924796 




1 


915453 


5 


1 


918607 


5 


1 


921738 


5 


1 


924848 


5 


. 2 


915505 


11 


2 


918659 


11 


2 


921790 


10 


2 


924899 


10 


3 


915558 


16 


3 


918712 


16 


3 


921842 


16 


3 


924951 


15 


4 


915611 


21 


4 


918764 


21 


4 


921894 


21 


4 


925002 


21 


5 


915664 


27 


5 


918816 


26 


5 


921946 


26 


5 


925054 


26 


6 


915716 


32 


6 


918869 


31 


6 


921998 


31 


6 


925106 


31 


7 


915769 


37 


7 


918921 


37 


7 


922050 


36 


7 


925157 


36 


8 


915822 


42 


8 


918973 


42 


8 


922102 


42 


8 


925209 


41 


9 


915874 


48 


9 


919026 


47 


9 


922154 


47 


9 


925260 


46 


8240 


915927 




8300 


95-9078 




8360 


922206 




8420 


925312 




1 


915980 


5 


1 


919130 


5 


1 1 


922258 


5 


1 


925364 


5 


2 


916033 


11 


2 


919183 


11 


i 2 


922310 


10 


2 


925415 


10 


3 


916085 


16 


3 


9192^5 


16 


1 3 


922362 


16 


3 


925467 


15 


4 


916138 


21 


4 


919287 


21 


4 


922414 


21 


4 


925518 


21 


5 


916191 


27 


5 


919340 


26 


5 


922466 


26 


5 


925570 


26 


6 


916243 


32 


6 


919392 


31 


6 


922518 


31 


6 


925621 


31 


7 


916296 


37 


7 


919444 


37 


7 


922570 


36 


7 


926673 ! 36 


8 


916349 


42 


8 


919496 


42 


I 8 


922622 


42 


8 


925724 1 41 


9 


916401 


48 


9 1 919549 


47 


i ' 9 


922674 


47 


9 


925776' 46 


8250 


916454 




8310 919601 




8370 


922725 




8430 


925828 


1 


916507 


5 


1 


919653 


5 


1 


922777 


5 


1 


925879 5 


2 


916559 


11 


2 


919705 


11 


2 


922829 


10 


2 


925931 ! 10 


3 


916612 


16 


3 


919758 


16 


3 


922881 


16 


3 


925982 I 15 


4 


916664 


21 


4 ' 919810 


21 


! 4 


922933 


21 


4 


926034' 21 


5 


916717 


26 


5 919862 


26 


1 5 


922985 


26 


5 


926085 26 


6 


916770 


31 


6 919914 


31 


6 


923037 


31 


6 


926137 3] 


7 


916822 


37 


7 919967 


37 


7 


923088 


36 


7 


926188 36 


8 


916875 


42 


'8 920019 


42 


8 


923140 


42 


8 


926239 1 41 


9 


916927 


47 


9 920071 


47 


9 923192 


47 

1 


9 926291 ' 46 



34 



LOGARITHMS OF NUMBERS. 



No. Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


1 

8440 1 926342 




8500 


929419 




8560 


932474 




8620 


935507 




1 j 926394 


5 


1 


929470 


5 


1 


932524 


5 


1 


935558 


5 


2 1 926445 


10 


2 


929521 


10 


2 


932575 


10 


2 


935608 


10 


8 1 926497 


15 


3 


929572 


15 


3 


932626 


15 


3 


935658 


15 


4 1 926548 


21 


4 


929623 


20 


4 


932677 


20 


4 


935709 


20 


5 926600 


26 


5 


929674 


26 


5 


932727 


25 


5 


935759 


25 


6 926651 


31 


6 


929725 


31 


6 


932778 


30 


6 


935809 


30 


7 926702 


36 


7 


929776 


36 


7 


932829 


35 


7- 


935860 


35 


8 1 926754 


41 


8 


929827 


41 


8 


932879 


40 


8 


935910 


40 


9 


926805 


46 


9 


929878 


46 


9 


932930 


45 


9 


935960 


45 


8450 


926857 




8510 


929930 




8570 


932981 




8630 


93601 1 




1 


926908 


5 


1 


929981 


5 


1 


933031 


5 


1 


936061 


5 


2 


926959 


10 


2 


930032 


10 


2 


933082 


10 


2 


936111 


10 


3 


927011 


15 


3 


930083 


15 


3 


933133 


15 


3 


936162 


15 


4 


927062 


21 


4 


930134 


20 


4 


933183 


20 


4 


936212 


20 


5 


927114 


26 


5 


930185 


26 


5 


933234 


25 


5 


936262 


25 


6 


927165 


31 


6 


930236 


31 


6 


933285 


30 


6 


936313 


80 


7 


927216 


36 


7 


930287 


36 


7 


933335 


35 


7 


936363 


35 


8 


927268 


41 


8 


930338 


41 


8 


933386 


40 


8 


936413 


40 


9 


927319 


46 


9 


930389 


46 


9 


933437 


45 


9 


936463 


45 


8460 


927370 




8520 


930440 




8580 


933487 




8640 


936514 




1 


927422 


5 


1 


930491 


5 1 


1 


933538 


5 


1 


936564 


5 


2 


927473 


10 


2 


930541 


10 


2 


933588 


10 


2 


936614 


10 


3 


927524 


15 


3 


930592 


15 


3 


933639 


15 


3 


936664 


15 


4 


927576 


21 


4 


930643 


20 


4 


933690 


20 


4 


936715 


20 


5 


927627 


26 


5 


930694 


25 


5 


933740 


25 


5 


936765 


25 


6 


927678 


31 


6 


930745 


31 


6 


933791 


30 


6 


936815 


30 


7 


927730 


36 


7 


930796 


36 


_ 


933841 


35 


7 


936865 


35 


8 


927781 


41 


8 


930847 


41 


8 


933892 


40 


8 


936916 


40 


9 


927832 


46 


9 


930898 


46 


9 


933943 


45 


9 


936966 


45 


8470 


927883 




8530 


930949 




8590 


933993 




8650 


937016 




1 


927935 


5 


1 


931000 


5 i 


1 


934044 


5 


1 


937066 


5 


2 


927986 


10 


2 


931051 


10 i 


2 


934094 


10 


2 


937116 


10 


3 


928037 


15 


3 


931102 


15 1 


3 


934145 


15 


3 


937167 


15 


4 


928088 


21 


4 


931153 


20 


4 


934195 


20 


4 


937217 


20 


5 


928140 


26 


5 


931203 


25 


5 


934246 


25 


5 


937267 


25 


6 


928191 


31 


6 


931254 


31 


6 


934296 


30 


6 


937317 


80 


7 


928242 


36 


7 


931305 


36 


■ 7 


934347 


35 


7 


937367 


35 


8 


928293 


41 


8 


931356 


41 


8 


934397 


40 


8 


937418 


40 


9 


928345 


46 


9 


931407 


46 


9 


934448 


45 


9 


937468 


45 


8480 


928396 




8540 


931458 




8600 


934498 




8660 


937518 




1 


928447 


5 


1 


931509 


5 


1 


934549 


5 


1 


937568 


5 


2 


928498 


10 


2 


931560 


10 


2 


934599 


10 


2 


937618 


10 


3 


928549 


15 


3 


931610 


15 


3 


934650 


15 


3 


937668 


15 


4 


928601 


21 


4 


931661 


20 


4 


934700 


20 


4 


937718 


20 i 


5 


928652 


26 


5 


931712 


25 


5 


934751 


25 


5 


937769 


25 1 


6 


928703 


31 


6 


931763 


31 


6 


934801 


30 


6 


937819 


30 


7 


928754 


36 


7 


931814 


36 


7 


934852 


35 


7 


937869 


35 


8 


928805 


41 


8 


931864 


41 


8 


934902 


40 


8 


937919 


40 


9 


928856 


46 


9 


931915 


46 


9 


934953 


45 


9 


937969 


45 


8490 


928908 




8550 


931966 




8610 


935003 




8670 


938019 




1 1 928959 


5 


1 


932017 


5 


1 


935054 


5 


1 


938069 


5 


2 i 929010 


10 


2 


932068 


10 


2 


935104 


10 


2 


938119 


10 


3 


929061 


15 


3 


932118 


15 


3 


935154 


15 


3 


938169 


15 


4 


929112 


20 


4 


932169 


20 


4 


935205 


20 


4 


938219 


20 


5 


929163 


26 


5 


932220 


25 


5 


935255 


25 


5 


938269 


25 


6 


929214 


31 


6 


932271 


30 


6 


935306 


30 


6 


938319 


30 


7 


929266 


36 


7 


932321 


35 


7 


935356 


35 


7 


938370 


85 


8 


929317 


41 


8 


932372 


40 


8 


935406 


40 


8 


938420 


40 1 


9 


929368 


46 


9 


932423 


45 


9 


935457 ! 45 | 


9 


938470 


4.j 



LOGARITHMS OF NUMBERS. 



35 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


8680 


938520 




8740 


941511 




8800 


944483 




8860 


947434 




1 


938570 


5 


1 


941561 


5 


1 


944532 


5 


1 


947483 


5 


2 


938620 


10 


2 


941611 


10 


2 


944581 


10 


2 


947532 


10 1 


3 


938670 


15 


3 


941660 


15 


3 


944631 


15 


3 


947581 


15 j 


4 


938720 


20 


4 


941710 


20 


4 


944680 


20 


4 


947630 


20 


5 


938770 


25 


5 


941760 


25 


5 


944729 


25 


5 


947679 


25 1 


6 


938820 


30 


6 


941809 


30 


6 


944779 


30 


6 


947728 


29 1 


7 


938870 


35 


7 


941859 


35 


7 


944828 


35 


7 


947777 


34 


8 


938920 


40 


8 


941909 


40 


8 


944877 


40 


8 


947826 


39 1 


9 


938970 


45 


9 


941958 


45 


9 


944927 


45 


9 


947875 


44 i 


8690 


939020 




8750 


942008 




8810 


944976 




:8870 


947924 




1 


939070 


5 


1 


942058 


5 


1 


945025 


5 


1 


947973 


5 


2 


939120 


10 


2 


942107 


10 i 


2 


945074 


10 


2 


948021 


10 


3 


939170 


15 


3 


942157 


15 1 


3 


945124 


15 


3 


948070 


15 


4 


939220 


20 


4 


942206 


20 \ 


4 


945173 


20 


4 


948119 


20 i 


5 


939270 


25 


5 


942256 


25 


5 


945222 


25 


5 


948168 


25 


6 


939319 


30 


6 


942306 


30 ^ 


6 


945272 


30 


6 


948217 


29 


7 


939369 


35 


7 


942355 


35 : 


7 


945321 


35 


1 7 


948266 


34 


8 


939419 


40 


8 


942405 


40 


8 


945370 


40 


1 8 


948315 


39 


9 


939469 


45 


9 


942454 


45 


9 


945419 


45 


1 9 


948364 


44 


8700 


939519 




8760 


942504 




8820 


945469 




8880 


948413 




1 


939569 


5 


1 


942554 


5 


1 


945518 


5 


1 


948462 


5 


2 


939619 


10 


2 


942603 


10 


2 


945567 


10 


2 


948511 


10 1 


3 


939669 


15 


3 


942653 


15 


3 


945616 


15 


3 


948560 


15 


4 


939719 


20 


4 


942702 


20 


4 


945665 


20 


4 


948608 


20 


•5 


939769 


25 


5 


942752 


25 


5 


945715 


25 


5 


948657 


25 


6 


939819 


30 


6 


942801 


30 


6 


945764 


29 


6 


948706 


.29 


7 


939868 


35 


7 


942851 


35 


7 


945813 


34 


! 7 


948755 


34 


8 


939918 


40 


8 


942900 


40 


8 


945862 


39 


8 


948804 


39 


9 


939968 


45 


9 


942950 


45 


9 


945911 


44 


9 


948853 


44 


8710 


940018 




8770 


943000 




8830 


945961 




8890 


948902 


1 


1 


940068 


5 


1 


943049 


5 


1 


946010 


5 


1 


948951 


5 


2 


940118 


10 


2 


943099 


10 


2 


946059 


10 


2 


948999 


10 


3 


940168 


15 


3 


943148 


15 


3 


946108 


15 , 


3 


949048 


15 


4 


940218 


20 


4 


943198 


20 


4 


946157 


20 1 


4 


949097 


20 


5 


940267 


25 


5 


943247 


25 


5 


946207 


25 


5 


949146 


25 


6 


940317 


30 


6 


943297 


30 


6 


946256 


29 j 


6 


949195 


29 


7 


940367 


35 


7 


943346 


35 


7 


946305 


34 i 


7 


949244 


34 


8 


940417 


40 


8 


943396 


40 


8 


946354 


39 1 


8 


949292 


39 


9 


940467 


45 


9 


943445 


45 


9 


946403 


44 


9 


949341 


44 


8720 


940516 




8780 


943494 




8840 


946452 




8900 


949390 




1 


940566 


5 


1 


943544 


5 


1 


946501 


5 


1 


949439 


5 


2 


940616 


10 


2 


943593 


10 


2 


946550 


10 


2 


949488 


10 


3 940666 


15 


3 


943643 


15 


3 


946600 


15 


3 


949536 


15 


4 


940716 


20 


4 


943692 


20 


4 


946649 


20 


4 


949585 


20 


5 


940765 


25 


5 


943742 


25 


5 


946698 


25 


5 


949634 


25 


6 


940815 


30 


6 


943791 


30 


6 


946747 


29 


6 


949683 


29 


7 


940865 


35 


7 


943841 


35 


7 


946796 


34 


7 


949731 


34 


8 


940915 


40 


8 


943890 


40 


8 


946845 


39 


8 


949780 


39 


9 


940964 


45 


9 


943939 


45 


9 


946894 


44 


9 


949829 


44 


8730 


941014 




8790 


943989 




8850 


946943 




8910 


949878 




1 


941064 


5 


1 


944088 


5 


1 


946992 


5 


1 


949926 


5 


2 


a41114 


10 


2 


944088 


10 


2 


947041 


10 


2 


949975 


10 


3 


941163 


15 


3 


944137 


15 


3 


947090 


15 


3 


950024 


15 


4 941213 


20 


4 


944186 


20 


4 


947139 


20 


4 


950073 


20 


5 941263 


25 


5 


944236 


25 


5 


947189 


25 


5 


950121 


25 


6 941313 


30 


6 


944285 


30 


6 


947238 


29 


6 


950170 


29 


7 941362 


35 


7 


944335 


35 


7 


947287 


34 


7 950219 


34 


8 941412 


40 


8 


944384 


40 


8 


947336 


39 


8 950267 


39 


9 941462 ' 45 


9 


944433 


45 

I 


9 


947385 


44 


9 950316 


44 



34 



86 



LOGARITHMS OF NUMBERS. 



No. 


1 


No. j Log. 


171 


No. 


Log. 


Prop. 1 
Part. ! 


No. 


Log. 


Prop. 
Part. 


8920 


950365 


1 


8980 


953276 




9040 


966168 


i 
1 


9100 


959041 




1 


950413 


5 


1 


953325 


5 


1 


966216 


5 1 


1 


959089 


5 


2 


950462 


10 


2 


963373 


10 


2 


956264 


10 1 




959137 


10 


3 


950511 


15 


3 


963421 


15 


3 


966312 


14 ! 


3 


959184 


14 


4 


950560 


19 


4 


953470 


19 


4 


956361 


19 


4 


969232 


19 


5 


950608 


24 


6 


953518 


24 


5 


956409 


24 


5 


959280 


24 


6 


950657 


29 


6 


953566 


29 


6 


966457 


29 ! 


6 


959328 


29 


7 


950705 


34 


7 


963615 


34 j 


7 


956505 


34 1 


7 


959375 


34 


8 


950754 


39 


8 


953663 


39 


8 


956563 


38 


8 


959423 


38 


9 


950803 


44 


9 


953711 


44 j 


9 


956601 


43 


9 


969471 


43 


8930 


950851 




8990 


953760 


1 


9050 


956649 


1 


9110 


969518 




1 


950900 


5 


1 


953808 


5 


1 


956697 


5 


1 


969666 


5 


2 


950949 


10 


2 


953856 


10 


2 


956745 


10 


2 


959614 


10 


3 


950997 


15 


3 


953905 


15 


3 


956792 


14 


3 


959661 


14 


4 


951046 


19 


4 


963963 


19 


4 


956840 


19 1 


4 


959709 


19 


6 


951095 


24 


5 


954001 


24 


5 


950888 


24 1 


5 


959757 


24 


6 


951143 


29 


6 


954049 


29 


6 


956936 


29 1 


6 


959804 


29 


7 


951192 


34 


7 


954098 


34 


7 


956984 


34 1 


7 


959852 


34 


8 


951240 


39 


8 


954146 


39 


8 


967032 


38 


8 


969900 


38 


9 


951289 


44 


9 


954194 


44 


9 


957080 


43 ; 


9 


959947 


43 


8940 


951337 




9000 


954242 




9060 


957128 




9120 


959996 




1 961386 


5 


1 


954291 


5 


1 


957176 


5 i 


1 


960042 


5 


2 


951435 


10 


2 


954339 


10 


2 


957224 


10 1 


2 


960090 


10 


3 


951483 


15 


3 


964387 


14 


3 


957272 


14 ! 


3 


960138 


14 


4 


951532 


19 


4 


964436 


19 1 


4 


957320 


19 i 


4 


960186 


19 


5 


951580 


24 


5 


964484 


24 


6 


957368 


24 ; 


5 


960233 


24 


6 


951629 


29 


6 


964632 


29 ^ 


6 


957416 


29 , 


6 


960280 


28 


7 


951677 


34 


7 


954580 


34 


7 


967464 


34 i 


7 


960328 


33 


8 


951726 


39 


8 


954628 


38 


8 


967511 


38 i 


8 


960376 


38 i 


9 


951774 


44 


9 


954677 


43 


9 


967559 


43 : 


9 


960423 


43 


8950 


951823 




9010 


964725 




9070 


957607 




9130 


960471 




1 


951872 


5 


1 


954773 


5 


1 


967665 


S i 


1 


960618 


5 


2 


951920 


10 


2 


954821 


10 


2 


957703 


10 


2 


960566 


10 


3 


951969 


16 


3 


954869 


14 


3 


957761 


14 


3 


960613 


14 


4 


952017 


19 


i 4 


964918 


19 


4 


957799 


19 


4 


960661 


19 , 


5 


962066 


24 


5 


964966 


24 


5 


957847 


24 ! 


5 


960709 


24 


6 


952114 


29 


6 


955014 


29 


6 


957894 


29 


6 


960756 


28 


7 


952163 


34 


7 


956062 


34 


7 


957942 


34 


7 


960804 


33 


8 


952211 


39 


8 


966110 


38 


8 


957990 


38 


8 


960851 


38 


9 


952259 


44 


9 


955158 


43 


9 


958038 


43 


9 


960899 


43 


8960 


952308 




9020 


956206 




9080 


958086 




9140 


960946 




1 


962356 


5 


1 


955265 


5 


1 


958134 


5 ' 


1 


960994 


6 


2 


952405 


10 


2 


955303 


10 


2 


958181 


10 i 


2 


961041 


10 


3 


952453 


16 


3 


965351 


14 


3 


958229 


14 1 


3 


961089 


14 


4 


962502 


19 


4 


956399 


19 


4 


958277 


19 ! 


4 


961136 


19 


5 


952550 


24 


5 


955447 


24 


5 


958325 


24 1 
29 ! 


5 


961184 


24 


6 


952699 


29 


6 


966495 


29 


6 


958373 


6 


961231 


28 


7 


962647 


34 


7 


965543 


34 


. 7 


958420 


34 : 


— 


961279 


33 


8 


952696 


39 


8 


955592 


38 


8 


958468 


38 1 


8 


961326 


38 


9 


962744 


44 


9 


955640 


43 


9 


958616 


43 


9 


961374 


43 


8970 


962792 




9030 


955688 




9090 


968564 




9150 


961421 




1 


962841 


5 


1 


955736 


5 


1 


958612 


5 i 


1 


961469 


5 


2 


962889 


1» 


2 


955784 


10 


2 


968659 


10 1 


2 


961516 


10 


3 


952938 


3 


956832 


14 


3 


958707 


14 1 


3 


961563 


14 


4 


952986 


19 


4 


966880 


19 


4 


958765 


19 


4 


961611 


19 


5 


953034 i 24 


5 


965928 


24 


5 


958803 


24 : 


5 


961658 


24 


6 


953083 1 29 


6 


955976 


29 


6 


958860 


29 ! 


6 


961706 


28 


7 


953131 


34 


7 


966024 


34 


7 


958898 


34 j 


7 


961753 


33 


8 


953180 


39 


8 


956072 


38 


8 


968946 


38 


8 


961801 


38 


9 


953228 


44 


9 


966120 


43 


9 


968994 


43 1 


9 


961848 


43 



LOGARITHMS OF NUMBERS. 



37 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


9160 


961895 




9220 


964731 




9280 


967548 




9340 


970347 




1 


961943 


5 


1 


964778 


5 


1 


967595 


5 


1 


970393 


5 


2 


961990 


10 


2 


964825 


9 


2 


967642 


9 


2 


970440 


9 


3 


962088 


14 


3 


964872 


14 


3 


967688 


14 


3 


970486 


14 


4 


962085 


19 


4 


964919 


19 


4 


967735 


19 


4 


970533 


19 


6 


962132 


24 


5 


964966 


24 


5, 


.967782 


28 


5 


970579 


23 


6 


962180 


28 


6 


965013 


28 


6 


967829 


28 


6 


970626 


28 


7 


962227 


33 


7 


965060 


33 


7 


967875 


33 


7 


970672 33 | 


8 


962275 


38 


8 


965108 


38 


8 


967922 


38 


8 


970719 


37 


9 


962322 


43 


9 


965155 


42 


9 


967969 


42 1 


9 


970765 


42 


9170 


962369 




9230 


965202 




9290 


968016 




9350 


970812 




1 


962417 


5 


1 


965249 


5 


1 


968062 


5 


1 


970858 


5 


2 


962464 


9 


2 


965296 


9 


2 


968109 


9 


2 


970904 


9 


3 


962511 


14 


3 


965343 


14 


3 


968156 


14 


3 


970951 


14 


4 


962559 


19 


4 


965390 


19 


4 


968203 


19 


4 


970997 


19 


5 


962606 


24 


5 


965437 


24 


5 


968249 


23 i 


5 


971044 


23 


6 


962653 


28 


6 


965484 


28 


6 


968296 


28 


6 


971090 


28 


7 


962701 


33 


7 


965531 


33 


7 


968343 


33 


7 


971137 


33 


8 


962748 


38 


8 


965578 


38 


8 


968389 


38 


8 


971183 37 1 


9 


962795 


42 


9 


965625 


42 


9 


968436 


42 


9 


971229 


42 


9180 


962843 




9240 


965672 




9300 


968483 


1 


9360 


971276 




1 


962890 


5 


1 


965719 


5 


1 


968530 


5 


1 


971322 


5 


2 


962937 


9 


2 


965766 


9 


2 


968576 


9 


2 


971369 


9 


3 


962985 


14 


3 


965813 


14 


3 


968623 


14 


3 


971415 


14 


4 


963032 


19 


4 


965860 


19 


4 


968670 


19 


4 


971461 


19 


5 


963079 


24 


5 


965907 


24 


5 


968716 


23 


5 


971508 


23 


6 


963126 


28 


6 


■965954 


28 


6 


968763 


28 


6 


971554 


28 


7 


963174 


33 


7 


966001 


33 


7 


968810 


33 


7 


971600 


33 


8 


963221 


38 


8 


966048 


38 


8 


968856 


37 


8 


971647 


37 


9 


963268 


42 


9 


966095 


42 


9 


968903 


42 


9 


971693 


42 


9190 


963315 




9250 


966142 




9310 


968950 




9370 


971740 




1 


963363 


5 


1 


966189 


5 


1 


968996 


5 


1 


971786 


5 


2 


963410 


9 


2 


966236 


9 


2 


969043 


9 


2 


971882 


9 


3 


963457 


14 


3 


966283 


14 


3 


969090 


14 


3 


971879 


14 


4 


963504 


19 


4 


966329 


19 


4 


969136 


19 


4 


971925 


19 


5 


963552 


24 


5 


966376 


24 


5 


969183 


23 


5 


971971 


23 


6 


963599 


28 


6 


966423 


28 


6 


969229 


28 


6 


972018 


28 


7 


963646 


33 


7 


966470 


33 


7 


969276 


33 


7 


972064 


33 


8 


963693 


38 


8 


966517 


38 


8 


969323 


37 


8 


972110 


37 


9 


963741 


42 


9 


966564 


42 


9 


969369 


42 


9 


972156 


42 


9200 


963788 




9260 


966611 




9320 


969416 




9380 


972203 




1 


963835 


5 


1 


966658 


5 


1 


969462 


5 


1 


972249 


5 


2 


963882 


9 


2 


966705 


9 


2 


969509 


9 


2 


972295 


9 


3 


963929 


14 


3 


9G6752 


14 


3 


969556 


14 


3 


972342 


14 


4 


963977 


19 


4 


966798 


19 


4 


969602 


19 


4 


972388 


18 


5 


964024 


24 


5 


966845 


24 


5 


969649 


23 


5 


972484 


23 


6 


964071 


28 


6 


966892 


28 


6 


969695 


28 


6 


972480 


28 


7 


964118 


33 


7 


966939 


33 


7 


969742 


33 


7 


972527 


32 


8 


964165 


38 


8 


966986 


38 


8 


969788 


37 


8 


972573 


37 


9 


964212 


42 


9 


967033 


42 


9 


969835 


42 


9 


972619 


41 


9210 


964260 




9270 


967080 




9330 


969882 




9390 


972666 




1 


964307 


5 


1 


967127 


5 


1 


969928 


5 


1 


972712 


5 


2 


964354 


9 


2 


967173 


9 


2 


969975 


9 


2 


972758 


9 


3 


964401 


14 


3 


967220 


14 


3 


970021 


14 


3 


972804 


14 


4 


964448 


19 


4 


967267 


19 


4 


970068 


19 


4 


972851 


18 


5 


964495 


24 


5 


967314 


24 


5 


970114 


23 


6 


972897 


23 


6 


964542 


28 


6 


967361 


28 


6 


970161 


28 


6 


972943 


28 


7 


964590 


33 


7 


967408 


33 


7 


970207 


33 


7 


972989 


32 


8 


964637 


38 


8 


967454 


38 


8 


970254 


37 


8 


973035 


37 


9 


964684 


42 


9 


967501 


42 


9 


970300 


42 


9 


973082 


41 



38 



LOGARITHMS OF NUMBERS. 



Nd. 1 Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 

Part. 


No. 


Log. 


Prop. 1 
Part, j 


No. 


Log. 


Prop. 
Part. 


9i00 973128 




9460 


975891 




9520 1 978637 




9580 


981365 




1 1 973174 


5 


1 


975937 


5 


1 1 978683 


5 


1 


981411 


5 


2 i 973220 


9 


2 


975983 


9 


2 


978728 


9 


2 


981456 


9 


3 1 973266 


14 


3 


976029 


14 


3 


978774 


14 


3 981501 


14 


4 1 973313 


18 


4 


976075 


18 


4 


978819 


18 


4 1 981547 


18 


5 973359 


23 


5 


9761^1 


23 


5 


978865 


23 


5 ' 981592 


23 


6 1 973405 


28 1 


6 


976166 


28 


6 


978911 


27 


6 ! 981637 


27 


7 ! 973451 


32 ! 


7 


976212 


32 


7 


978956 


32 


7 i 981683 


32 1 


8 : 973497 


37 i 


8 


976258 


37 


8 


979002 


36 


8 j 981728 


36 


9 1 973543 


41 1 


9 


976304 


41 


9 


979047 


41 


9 


981773 


41 


9410 


973590 




9470 


976350 




9530 


979093 




9590 


981819 




1 


973636 


5 


1 


976396 


5 


1 


979138 


5 


1 


981864 


6 


2 


973682 


9 


2 


976442 


9 


2 


979184 


9 


2 


981909 


9 


3 


973728 


14 


3 


976487 


14 


3 


979230 


14 


3 


981954 


14 


4 


973774 


18 


4 


976533 


18 


4 


979275 


18 


4 


982000 


18 


5 


973820 


23 


5 


976579 


23 


5 


979321 


23 


5 


982045 


23 


6 1 973866 


28 


6 


976625 


28 


6 


979366 


27 


6 


982090 


27 


7 i 973913 


32 


7 


976671 


32 


7 


979412 


32 


7 


982135 


32 


8 973959 


37 


8 


976717 


37 


8 


979457 


36 


8 


982181 


36 


9 974005 


41 


9. 


976762 


41 


9 


979503 


41 


9 


982226 


41 


9420 974051 




9480 


976808 




9540 


979548 




9600 


982271 




1 1 974097 


5 


1 


976854 


5 


1 


979594 


5 


1 


982316 


5 


2 


974143 


9 


2 


976900 


9 


2 


979639 


9 


2 


982362 


9 


3 


974189 


14 


3 


976946 


14 


8 


979685 


14 


8 


982407 14 


4 


974235 


18 


4 


976991 


18 


4 


979730 


18 


4 


982452 18 


5 


974281 


23 


5 


977037 


23 


5 


979776 


23 


5 


982497 


23 


6 


974327 


28 


6 


977083 


27 


6 


979821 


27 


6 


982543 


27 


7 


974373 


32 


7 


977129 


32 


7 


979867 


32 


7 


982588 


32 


8 


974420 


37 


8 


977175 


37 


8 


979912 


36 


8 


982633 


36 


9 ! 974466 


41 


9 


977220 


41 


9 


979958 


41 


9 


982678 


41 


9430 1 974512 




9490 


977266 




9550 


980003 




9610 


982723 




1 974558 


5 


» 1 


977312 


5 


1 


980049 


5 


1 


982769 


5 


2 


974604 


9 


2 


977358 


9 


2 


980094 


9 


2 


982814 


9 


3 


974650 


14 


3 


977403 


14 


3 


980140 


14 


3 


982859 


14 


4 


974696 


18 


4 


977449 


18 


4 


980185 


18 


4 


982904 


18 


5 


974742 


23 


5 


977495 


23 


5 


980231 


23 


5 


982949 


23 


6 


974788 


28 


6 


977541 


27 


6 


980276 


27 


6 


982994 


27 


7 


974834 


32 


7 


977586 


32 


7 


980322 


32 


7 


983040 


32 


8 


974880 


37 


8 


977632 


37 


8 


980367 


36 


8 


983085 


36 


9 


974926 


41 


9 


977678 


41 


9 


980412 


41 


9 


983130 


41 


9440 


974972 




9500 


977724 




9560 


980458 




9620 


983175 




1 


975018 


5 


1 


977769 


5 


1 


980503 


5 


1 


983220 


5 


2 


975064 


9 


2 


977815 


9 


2 


980549 


9 


2 


983265 


9 


3 


975110 


14 


3 


977861 


14 


3 


980594 


14 


3 


983310 


14 


4 


975156 


18 


1 4 


977906 


18 


4 


980640 


18 


4 


983356 


18 


5 


975202 


23 


i 5 


977952 


23 


5 


980685 


23 


5 


983401 


23 


6 


975248 


28 


6 


977998 


27 


6 


^80730 


27 


6 


983446 


27 


7 


975294 


32 


7 


978043 


32 


7 


980776 


32 


7 


983491 


32 


8 


975340 


37 


8 


978089 


37 


8 


980821 


36 


8 


983536 


36 


9 


975386 


41 


9 


978135 


41 


9 


980867 


41 


9 


983581 


41 


9450 


975432 




9510 


978180 




9570 


980912 




9630 


983626 




1 


975478 


5 


1 


978226 


5 


1 


980957 


5 


1 


983671 


5 


2 


975524 


9 


2 


978272 


9 


2 


981003 


9 


2 


983716 


9 


3 


975570 


14 


3 


978317 


14 


3 


981048 


14 


3 


983762 


14 


4 


975616 


18 


4 


978363 


18 


4 


981093 


18 


4 


983807 


18 


5 


975661 


23 


5 


978409 


23 


5 


981139 


23 


5 


983852 


23 


6 


975707 


28 


6 


978454 


27 


« 


981184 


27 


6 


983897 


27 


7 


975753 


32 


7 


978500 


32 


7 


981229 


32 


7 


983942 


32 


8 


975799 


37 


8 


978546 


37 


i 8 


981275 


36 


8 ! 983987 


36 


9 


975845 


41 


9 


978591 


41 


1 ^ 


981320 


41 


9 1 984032 


41 



LOGARITHMS OF NUMBERS. 



39 



No. 


Lug. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Par. 


No. 


Log 


Prop. 
Pare. 


1 No. i Log. i^f^P- 


9640 


984077 




9700 986772 




9760 


989450 




! 9820 992111 i 


1 


984122 


5 


1 ! 986816 j 4 


1 


989494 


4 


! 1 ' 992156 ! 4 


2 


984167 


9 


2 ; 986861 9 


2 


989539 


9 


2 992200! 9 


3 


984212 


14 


3 986906 1 13 | 


3 


989583 


13 


1 3 992244 : 13 


4 


984257 18 j 


4 : 986951 ! 18 ! 


4 


989628 


18 


4 992288: 18 


6 


984302 


23 


5 ' 986995 1 22 j 


5 


989672 


22 


5 ' 992333 \ 22 


6 


984347 


27 


6 987040 1 27 i 


6 


989717 


27 


6 992377 1 26 


7 


984392 


32 


7 i 987085 i 31 ! 


7 


989761 


31 


! 7 i 992421 31 


8 


984437 


36 


8 1 987130 


36 ' 


8 


989806 


36 


8 1992465 35 


9 


984482 


41 


9 i 987174 


40 1 


9 j 989860 


40 


9 1 992509 1 40 


9650 


984527 




9710 987219 


i 


9770 989895 




9830 ; 992553 


1 


1 


984572 


5 


1 ' 987264 


^i 


1 


989939 


4 


1 i 992598 


4 1 


2 


984617 


9 


2 


987309 


9 1 


2 


989983 


9 


2 i 992642 


9 


3 


984662 


14 


3 


987353 


13 ! 


3 


990028 


13 


3 i 992686 


13 


4 


984707 


18 


4 


987398 


18 i 


4 


990072 


18 


, 4 992730 


18 


5 


984752 


23 


5 


987443 


.22 ' 


5 


990117 


22 


5 1 992774 


22 


6 


984797 


27 


6 


987487 


2' i 


6 


990161 


27 


6 : 992818 


26 


7 


984842 


32 


7 987532 


31 


7 


990206 


31 


7 i 992863 1 31 


8 


984887 


36 


8 987577 


36 1 


8 


990250 


36 


8,992907 1 35 


9 


984932 


41 


9 987622 


40 1 


9 


990294 


40 


9 992951 i 40 


9660 


984977 




9720 1 987666 


1 


9780 


990339 




9840 ! 992995 | 


1 


985022 


5 


1 1 987711 


4 


1 


990383 


4 


1 993039 


4 


2 


985067 


9 


2 1 987756 


9 


2 


990428 


9 


2 993083 


9 


3 


985112 


14 


3 1 987800 


13 


3 


990472 


13 


3 993127 


13 


4 


986157 


18 


4 ! 987845 


18 


4 


990516 


18 


4 1 993172 


18 


5 


985202 


23 


5 987890 


22 j 


6 


990561 


22 


5 1 993216 


22 


6 


985247 


27 


6 987934 


27 1 


6 


990605 


27 


6 ! 993260 


26 


7 


985292 


32 


7 987979 


31 i 


7 


990650 


31 


7 993304 


31 


8 


985337 


36 


8 988024 


36 i 


8 990694 


36 


8 993348 


35 


9 


985382 


41 


9 1 988068 


40 ! 


9 1 990738 


40 


9 993392 


40 


9670 


985426 




9730 , 988113 ! 1 


9790 i 990783 




9850 993436 




1 


985471 


4 


1 ! 988157 


4 


1 1 990827 


4 


1 993480 


4 


2 


985516 


9 


2 988202 


9 


2 1 990871 


9 


2 993524 


9 


3 


985561 


13 


3 i 988247 


13 


3 


990916 


13 


3 1 993568 


13 


4 985606 


18 


4 ] 988291 


18 


4 


990960 


18 


4 ' 993613 


18 


5 


985651 


22 


6 


988336 


22 


5 


991004 


22 


5 993657 


22 


6 


985696 


27 


6 


988381 


27 


6 


991049 


27 


6 ; 993701 


26 


7 


985741 


31 


7 


988425 


31 


7 


991093 


31 


7 {993745 31 i 


8 


985786 


36 


8 


988470 


36 


8 


991137 


36 


8 i 993789 j 35 | 


9 


985830 


40 


9 


988514 


40 


9 


991182 


40 


9 


993838 j 40 | 


9680 


985875 




9740 


988559 




9800 


991226 




9860 


993877 


1 


1 


985920 


4 


1 


988603 


4 


1 


991270 


4 


1 


993921 


4 


2 


985965 


9 


2 


988648 


9 


2 


991315 


9 


2 i 993965 


9 


3 


986010 


13 


3 


988693 


13 


3 


991359 


13 


3 994009 


13 


4 


986055 


18 


4 


988737 


18 


4 


991403 


18 


4 j 994053 


18 


5 


986100 


22 


5 


988782 


22 


6 


991448 


22 


5 ! 994097 


22 


6 


986144 


27 


6 


988826 


27 


6 


991492 


27 


6 1 994141 


26 


7 


986189 


31 


7 


988871 


31 


7 


991536 


31 


7 ! 994185 


31 1 


8 


986234 


36 


8 


988915 


36 


8 


991580 


36 


8! 994229 1 35 


9 


986279 


40 


9 


988960 


40 


9 


991625 


40 


9 \ 994273 ' 40 


9690 


986324 




9750 


989005 




9810 


991669 




9870 


994317 1 


1 


986369 


4 


1 


989049 


4 


1 


991713 


4 


1 


994361 4 


2 ' 986413 


9 


2 


989094 


9 


2 


991767 


9 


2 


994405 


9 


3 986458 


13 


3 


989138 


13 


3 


991802 


13 


3 


994449 


13 


4 986503 


18 


4 


989183 


18 


4 


991846 


18 


4 


994493 


18 


6 ; 986548 


22 


5 


989227 


22 


5 


991890 


22 


5 


994537 


22 


6 1 986593 


27 


6 


989272 


27 


6 


991934 


27 


6 


994581 


26 


7 


986637 


31 


7 


989316 


31 


7 


991979 


31 


i 7 


994625 


31 


8 


986682 


36 


8 


989361 


36 


8 


992023 


36 


1 8 


994669 1 35 


9 


986727 


40 


9 


989405 


40 


9 992067 


40 


1 ^ 


994713] 40 



40 



LOGARITHMS OF NUMBERS. 



No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. 
Part. 


No. 


Log. 


Prop. I 
Part, j 


No. 


Log. 


IZl 


9880 


994757 




9910 


996074 




9940 


997386 




9970 


998695 


1 


1 


994801 


4 


1 


996117 


4 


1 


997430 


4 


1 


998739 


4 


2 


994845 


9 


2 


996161 


9 


2 


997474 


9 


2 


998782 


9 


3 


994889 


13 


3 


996205 


13 


3 


997517 


13 


3 


998826 


13 


4 


994933 


18 


4 


996249 


18 


4 


997561 


17 


4 


998869 


17 


5 


994977 


22 


5 


996293 


22 


5 


997605 


22 


5 


998913 


22 


6 


995021 


26 


6 


996336 


26 


6 


997648 


26 


6 


998956 


26 


7 


995064 


31 


7 


996380 


31 


7 


997692 


30 


7 


999000 


30 


8 


995108 


35 


8 


996424 


35 


8 


997736 


35 


8 


999043 


35 


9 


995152 


40 


9 


996468 


40 


9 


997779 


39 


9 


999087 


39 


9890 


995196 




9920 


996512 




9950 


997823 




9980 


999130 




1 


995240 


4 


1 


996555 


4 


1 


997867 


4 


1 


999174 


4 


2 


995284 


9 


2 


996599 


9 


2 


997910 


9 


2 


999218 


9 


3 


995328 


13 


3 


996643 


13 


3 


997954 


13 


3 


999261 


13 


4 


995372 


18 


4 


996687 


18 


4 


997998 


17 


4 


999305 


17 


5 


995416 


22 


5 


996730 


22 


.5 


998041 


22 


5 


999348 


22 


6 


995460 


26 


6 


996774 


26 


6 


998085 


26 


6 


999392 


26 


7 


995504 


31 


7 


996818 


31 


7 


998128 


30 


7 


999435 


30 


8 


995547 


35 


8 


996862 


35 


8 


998172 


35 


8 


999478 


35 


9 


995591 


40 


9 


996905 


40 


9 


998216 


39 


9 


999522 


39 


9900 


995635 




9930 


996949 




9960 


998259 




9990 


999565 




1 


995679 


4 


1 


996993 


4 


1 


998303 


4 


1 


999609 


4 


2 


995723 


9 


2 


997037 


9 


2 


998346 


9 


2 


999652 


9 


3 


995767 


13 


3 


997080 


13 


3 


998390 


13 


3 


999696 


13 


4 


995811 


18 


4 


997124 


18 


4 


998434 


17 


4 


999739 


17 


5 


995854 


22 


5 


997168 


22 


5 


998477 


22 


5 


999783 


22 


6 


995898 


26 


6 


997212 


26 


6 


998521 


26 


6 


999826 


26 


7 


995942 


31 


7 


997255 


31 


7 


998564 


30 


7 


999870 


30 


8 


995986 


35 


8 


997299 


35 


8 


998608 


35 


8 


999913 


35 


9 


996080 


40 


9 i 997343 


39 


9 


998652 


39 


9 


999957 


39 



Logarithms to 50 Decimal Places. 



0-00000000000000000000000000000000000000000000000000 
0-30102999566398119521373889472449302676818988146211 
0-47712125471966243729502790325511530920012886419069 
0-60205999132796239042747778944898605353637976292422 
0-69897000433601880478626110527550697323181011853789 
0-77815125038364363250876679797960833596831874565280 
0-845098040014256830712216258592636193483572396-32397 
0-90308998699194358564121668417347908030456964438633 
0-95424250943932487459005580651023061840025772838139 
1-00000000000000000000000000000000000000000000000000 
1-04139268515822504075019997124302424170670219046645 
1-07918124604762482772250569270410136273650862711491 
1-11394335230683676020650515794232843082972918838707 
1-1461280356782380259259551533171292202^5176227778607 
1-17609125905568124208128900853062228243193898272859 
1-20411998265592478085495557889797210707275952584843 
1-23044892137827392854016989432833703000756737842505 
1-25527250510330606980379470123472364516844760984350 
1-27875360095282896153633347575692931795112933739450 
1-30102999566398119521373889472449302676818988146211 
1-32221929473391926800724416184775150268370126051866 
1-34242268082220623596393886596751726847489207192856 
1-36172783601759287886777711225118954969751103433610 
1-38021124171160602293624458742859438950469850857702 
1-39794000867203760957252221055101394646362023707578 



LOGARITHMIC SINES, ETC. 



41 



SEG. 






Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


Cosine. 


' 






Infinite. 






Infinite. 


•000000 


10-000000 


60 


1 


6-463726 




3-536274 


6-463726 




13-536274 


-000000 


10 000000 


59 


2 


6-764756 


501717 


3-235244 j6-764756 


501717 


13-235244 


-000000 


10-000000 


58 


3:6-940847 


293485 


3-0591531 6-940847 


293485 


13-059153 


-000000 


10-000000 


57 


41 ,7-065786 


208231 


2-934214;!7-065786 


208231 


12-934214 


•000000 


10-000000 


56 


5|7-162696 


161517 


2-837304!i7-162696 


161517 


12-837304 


-000000 


10-000000 


55 


6|;7-241877 


131968 


2-758123! 7-241878 


131969 


12-758122 


•000001 


9-999999 


54 


7'i7-308824 


111578 


2-6911767-308825 


111578 


12-691175 


•000001 


9-999999 


53 


8|i7-366816 


96653 


2-633184' 7-366817 


96653 


12-633183 


•000001 


9-999999 


52 


9i|7-417968 


85254 


2-582032| 7-417970 


85254 


12-582030 


-000001 


9-999999 


51 


10|7-463726 


76262 


2-536274! 7-463727 


76263 


12-536273 


•000002 


9-999998 


50 


lllj7 -505118 


68988 


2-494882S7-505120 


68988 


12-494880 


•000002 


9-999998 


49 


12!|7-542906 


62981 


2-457094!j7-542909 


62981 


12-457091 


•000003 


9-999997 


48 


13!7-577668 


57936 


2-422332 


7-577672 


57937 


12-422328 


•000003 


9-999997 


47 


14 7-609853 


53641 


2-390147 


7-609857 


53642 


12-390143 


•000004 


9^999996 


46 


15 7-639816 


49938 


2-360184 


7-639820 


49939 


12-360180 


•000004 


9-999996 


45 


1<5 7-667845 


46714 


2-332155 


7-667849 


46715 


12-332151 


•000005 


9-999995 


44 


17.7-694173 


43881 


2-3058271 


7-694179 


43882 


12-305821 


•000005 


9-999995 


43 


18;i7-718997 


41372 


2-28l003'j7-719003 


41373 


12-280997 


•000006 


9-999994 


42 


19|7-742478 


39135 


2-257522il7 -742484 


39136 


12-257516 


-000007 


9-999993 


41 


207-764754 


37127 


2-235246I7-764761 


37128 


12-235239 


-000007 


9^999993 


40 


217-785943 


35315 


2-21405717-785951 


35315 


12-214049 


•000008 


9-999992 


39 


22 7-806146 


33672 


2-193854! 7-806155 


33673 


12-193845 


•000009 


9-999991 


38 


2317-825451 


32175 


2-174549ii7-825460 


32176 


12-174540 


•000010 


9^999990 


37 


24,7-843934 


30805 


2-156066! 7-843944 


30807 


12-156056 


•000011 


9-999989 


36 


2517-861662 


29547 


2-138338!7-861674 


29549 


12-138326 


•000011 


. 9^999989 


35 


2617-878695 


28388 


2-121305!i7-878708 


28390 


12-121292 


•000012 


9-999988 


34 


27 7-895085 


27317 


2-104915li7'895099 


27318 


12-104901 


-000013 


9-999987 


33 


28 7-910879 


26323 


2-089121 


7-910894 


26325 


12-089106 


•000014 


9-999986 


32 


29:7-926119 


25399 


2-073881 


7-926134 


25401 


12-073866 


•000015 


9-999985 


31 


30 7-940842 


24538 


2-059158 


7-940858 


24540 


12-059142 


•000017 9-999983 


30 


31 '7-955082 


23733 


2044918 7-955100 


23735 


12-044900 


•000018 9^999982 


29 


32^;7-968870 


22980 


2-031130 


7-968889 


22982 


12-031111 


•000019 9-999981 


28 


33i7-982233' 22273 


2-017767 


7-982253 


22275 


12-017747 


•OOOO2O1 9^999980 


27 


3417-995198 21608 


2-004802 


7-995219 


21610 


12-004781 


•000021 


9-999979 


26 


35i8-007787 20981- 


1-992213 


8-007809 


20983 


11-992191 


•000023 


9-999977 


25 


36 8-020021 20390 


i-979979 


8-020045 


20392 


11-979955 


•000024 


9-999976 


24 


37 8-031919 19831 


1-968081 


8-031945 


19833 


11-968055 


•000025 


9-999975 


23 


38 8-043501 19302 


1-956499 


8-043527 


19305 


11-956473 


•000027 


9-999973 


22 


39 8-054781 18801 


1-945219 


8-054809 


18803 


11-945191 


•000028 


9-999972 


21 


40 8-065776 18325 


1-934224 


8-065806 


18327 


11-934194 


•000029 


9-999971 


20 


418-076500 17872 


1-923500 


8-076531 


17875 


11-923469 


•000031 


9-999969 


19 


42 8-086965 17441 


1-913035 


8-086997 


17444 


11-913003 


•000032 


9-999968 


18 


43 8-097183 17031 


1-902817 


8-097217 


17034 


11-902783 


-000034 


9-999966 


17 


44 8-107167 16639 


1-892833 


8-107202 


16642 


11-892798 


•000036 


9-999964 


16 


45'8-116926 16265 


1-883074 


8-116963 


16268 


11-883037 


•000037 


9-999963 


15 


46 


8-126471 15908 


1-873529 


8-126510 


15911 


11-873490 


•000039 


9.-999961 


14 


47 


8-135810 15566 


1-864190 


8-135851 


15568 


11-864149 


•000041 


9-999959 


13 


48 


8-144953 15238 


l-855047!!8-144996 


15241 


11-855004 


•000042 


9-999958 


12 


49 


8-153907 14924 


1-846093! 8-153952 


14927 


11-846048 


•000044 


9^999956 


11 


50 


8-162681 14622 


1-837319 8-162727 


14625 


11-837273 


•000046 


9-999954 


10 


51 


8-171280 14333 


1-828720 8-171328 


14336 


11-828672 


•000048 


9-999952 


9 


52!>8-179713 14054 


1-820287,8-179763 


14057 


11-820237 


•000050 


9-999950 


8 


53';[8-187985 13786 


1-812015 18-188036 


13790 


11-811964 


•000052 


9^999948 


7 


54il8-196102 13529 


1-803898 !8-196156 


13532 


11-803844 


•000054 


9^999946 


6 


55 8-204070 13280 


1-795930 8-204126 


13284 


11-795874 


•000056 


9-999944 


5 


56 8-211895 13041 


1-788105 '8-211953 


13044 


11788047 


•000058 


9-999942 


4 


57 8-219581 12810 


1-780419 8-219641 


12814 


11780359 


. -000060 


9-999940 


3 


5818-227134 12587 


1772866:8-227195 


12591 


11-772805, -000062 


1 9-999938 


2 


5918-234557 12372 


1-765443 8-234621 


12376 


11-765379 


-000064 


9-999936 


1 


60,8-241855 12164 


1758145 8-241922 


! 12168 


11-758078 


-0000661 9-999934 





' il Cosine. 


Secant. C .tangent. 


! Tangent. 


Cosecant. 1 Sine. 



89 D£G. 



42 



LOGARITHMIC SINES, ETC. 



1 


DEC. 














■ 


1 Sine. 


^^,; Cosecant. |! Tangent. 


DifF. 
100" 


Cotangent, i Secant. 


Cosine. ; 


' 





8-241865 




1-758145 8-241921 




11-758079 -000066 


9-999934 


60 1 


1 


8-249033 


11963 


1-750967 8-249102 


11967 


11-750898-000068 


9-999932: 59 


2 


8-256094 


11768 


1-743906 8-256165 


11772 


11-743835-000071 


9-999929 58 


3 


:8-263042 


11580 


1-736958 8-263115 


11584 


11-736885 -000073 


9-999927. 57 


4 


8-269881 


11397 


1-730119 8-269956 


11402 


11-730044 -000075 


9-999925 


56 


5 8-276614 


11221 


1-723386 8-276691 


11225 


11-723309-000078 


9-999922 


55 


6 8-283243 


11050 1-716757 8-283323 


11054 


11-716677-000080 


9-999920 


64 


7 18-289773 


10883 


1-710227 8-289856 


10887 


11-710144 -000082 


9-999918: 


53 


8j 8-296207 


10722 


1-703793 8-296292 


10726 


11-703708-000085 


9-999915 


52 


9 8-302546 


10565 


1-697454 8-302634 


10570 


11-697366 -000087 


9-999913 


51 


10|8-308794 


10413 


1-691206 8-308884 


10418 


11-691116-000090 


9-999910 ■ 


50 


li;;8-314954 


10266 


1-685046 8-315046 


10270 


11-684954-000093 


9-999907 


49 i 


12j8-321027 


10122 


1-678973 8-321122 


10126 


11-678878: -000095 


9-999905 


48 


13ji8-3270l6 


9982 


1-672984 8-327114 


9987 


11-672886 ;-000098 


9-999902 


47 


1418-332924 


9847 


1-667076 8-333025 


9851 


11-666975 i-000101 


9-999899 


46 


15 8-338753 


9714 


1-661247 8-338856 


9719 


11-661144-000103 


9-999897 


45 1 


16,8-344504 


9586 


1-6-55496 8-344610 


9590 


11-655390-000106 


9-999894 44 


17l;8-350181 


9460 


1-649819 8-350289 


9465 


11-649711-000109 


9-999891 43 


1818-355783 


9338 


1-644217 8-355895 


9343 


11-6441051-000112 


9-999888 ,42 


19'8-361315 


9219 


■1-638685 8-361430 


9224 


11-638570-000115 


9-999885 41 


20 8-366777 


9103 


1-633223 8-366895 


9108 


11-633105-000118 


9-999882 ■ 


40 


21 8-372171 


8990 


1-627829 8-372292 


8995 


11-62770S -000121 


9-999879 


39 


22 8-377499 


8880 


1-622501 8-377622 


8885 


11-622378-000124 


9-999876 


38 


23 8-382762 


8772 


1-617238 8-382889 


8777 


11-017111 i-000127 


9-999873 


37 


24 8-387962 


8667 


1-612038 8-388092 


8672 


11-611908-000130 


9-999870 


36 


25 8-393101 


8564 


1-606899 8-393234 


8570 


11-606766-0001.33 


9-999867 


35 


26 8-398179 


846411-601821 8-398315 


8470 


11-601685-000136 


9-999864 


34 


27 8-403199 


8366 1-596801 8-408338 


8371 


11-596662 -000139 


9-999861 


33 


28 8-408161 


8271 1-591839 8-408304 


8276 


11-591696 ;-000142 


9-999858 


32 


29 8-413068 


8177 1-586932 8-413213 


8182 


11-586787.-000146 


9-999854 


31 


30:|8-417919 


8086 1-582081 8-418068 


8091 


11-581932,-000149 


9-999851 


30 


8118-422717 


7996 4-577283 8-42-2869 


8002 


11-5771311-0001-52 


9-999848 


29 


32 8-427462 


7909 l-572538!:8-427618 


7914 '11-572382 -000156 


9-999844 


28 


33 8-432156 


7823 1-567844 ;8-432315 


7829 11-567685 r000159 


9-999841 


27 


34 8-436800 


7740 1-563200 8-436962 


7745 11 -56.30.38 -000162 


9-999838 


26 


35 8-441394 


7657 ;1 -558606 8-441560 


7663 11-558440-000166 


9-999834 


25 


36 8-445941 


7577 1-554059 8-446110 


7583 11-553890 -000169 


9-999831 


'24 


37 8-450440 


7499 1-549560 8-450613 


7505 11-5493871-000173 


9-999827 


23 


38 8-454893 


7422 1-545107 8-455070 


7428 11-544930-000176 


9-999824 


22 


39 8-459301 


7346 1-540699 8-459481 


73-52 11-540.519: -000180 


9-999820 


21 


40 8-463665 


7273 1-536335 8-463849 


7279 11-536151-000184 


9-999816 


20 


41 8-467985 


7200 1-532015 8-468172 


7206 11-531828 i-000187 


9-999813 


19 


42 8-472263 


7129 1-527737 8-472454 


7135 11 -.527.5461 -000191 


9-999809 


18 


43 8-476498 


70601-523.502 8-476693 


7066:11-5233071-000195 


9-999805 


17 


44 8-480693 


6991 ;1-519307 8-480892 


6998 '11-519108^000199 


9-999801 


16 


45,8-484848 


6924 1-515152 8-485050 


' 6931 ill -5149.50 :-O00203 


9-999797 


15 


46 ;8 -488963 


6859 


1-511037 ■8-489170 


6865 11-510830; -000206 


9-999794 


14 


47 8-493040 


6794 


1-506960 8-493250 


6801 I11-.506750 '-000210 


9-999790 


13 


48 ■8-497078 


6731 


;1 -.502922 18-497293 


67.38 11 -.502707 -000214 


9-999786 


12 


49 8-501080 


6669 


1-498920,8-501298 


6676 11-498702 -000218 


9-999782 


|11 


50 8-505045 


6608 


1-494955,8-505267 


6615 11-494733 -000222 


9-999778 


|10 


51:8-508974 


6548 


1-491026,8-509200 


6.555 11-490800-000226 


9-999774 


9 


62i!8-512867 


' 6489 


1-4871.33 8-513098 


6496 11-486902-000231 


9-999769 


1 8 


53|8-516726 


: 6432 


1-483274 8-516961 


6439 11-483039 -000235 


9-999765 


1 7 


54 8-520551 


' 6375 


1-479449 8-520790 


6.382 n -479210 -000239 


9-999761 


6 1 


55118-524343 


6319 


1-47.5657 8-524586 


6326 11 •47-541 4 -000243 


^•999757 


\ 5 1 


56i!8-528102 


6264 


1-471898 8-528349 


6272 11-471651 -000247 


9-999753 


4 i 


57'8-531828 


6211 1-468172 8-53l'U80 


6218 11-467920 -0002-52 9-999748 


1 3 1 


58 8-535523 


6158 1-464477 8-535779 


6165 11-464221-000256 9-999744i| 2 i 


59; '8 -539185 


. 6106 1-460814 8-539447 


6113 11 -460)5311-000260 9-999740!.; 1 1 


i 60 8-542819 


6055 1-457181 8-513084 


6062 ;il-456916;'-000265 9-999735 


; 


i ' ^'-"'- 


secaiit. C-.caugent. 


Taut' . '.-^ecant. S ne. 





LOGARITHMIC SINES, ETC. 



43 



2 TiEQ. 



' 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diflf. 
100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


' 





8-542819 




1-457181 


8-543084 




11-456916 


-000265 




9-999735 


60" 


1 


« -546422 


6004 


1-453578 


8-546691 


6012 


11-453309 


-000269 


07 


9-999731 


59 


2 


8-549995 


5955 


1-450005 


8-550268 


5962 


11-449732 


-000274 


07 


9-999726 


58 


3 


8-553539 


5906 


1-446461 


8-553817 


5914 


11-440183 


-000278 


08 


9-999722 


57 


4 


8-557054 


5858 


1-442946 


8-557336l5866| 


11-442664 


-000283 


08 


9-999717 


56 


5 


8-560540 


5811 


1-439460 


8-560828 


5819 


11-439172 


-000287 


07 


9-999713 


55 


6 


8-563999 


5765 


1-436001 


8-564291 


5773 


11-435709 


-000292 


08 


9-999708 


54 


7 


8-567431 


5719 


1-4-32569 


8-567727 


5727 


11-432273 


-000296 


07 


9-999704 


53 


8 


8-570836 


5674 


1-429164 


8-571137 


5682 


11-428863 


-000301 


08 


9-999699 


52 


9 


8-574214 


5630 


1-425786 


8-574520 


5638 


11-425480 


-000306 


08 


9-999694 


51 


10 


8-577566 


5587 


1-422434 


8-577877 


5595 


11-422123 


-000311 


08 


9-999689 


50 


11 


8-580892 


5544 


1-419108 


8-581208 


5552 


11-418792 


-000315 


07 


9-999685 


49 


12 


8-584193 


5502 


1-415807 


8-584514 


5510 


11-415486 


-000320 


08 


9-999680 


48 


13 


8-587469 


5460 


1-412531 


8-587795 


5468 


11-412205 


-000325 


08 


9-99t.675i 


47 


14 


8-590721 


5419 


1-409279 


8-591051 


5427 


11-408949 


-000330 


08 


9-999670 


46 


151 


8-593948 


5379 


1-406052 


8-594283 


5387 


11-405717 


•000335 


08 


9-9^9665 


45 


Ibl 


8-597152 


5339 


1-402848 


8-5974925347 


11-402508 


-000340 


08 


9-999660 


44 


171 


8-600332 


5300 


1-399668 


8-600677 5308 


11-399323 


-000345 


08 


9-999655 


43 


18! 


8-603489 


5261 


1-396511 


8-603839i5270 


11-396161 


-000350 


08 


9-999650 


42 


19 


8-606623 


5223 


1-393377 


8-60697815232 


11-393022 


•000355 


08 


9-999645 


4i 


20 


8-609734 


5186 


1-390266 


8-01009415194 


11-389906 


-000360 


08 


9-999640 


40 


21 


8-612823 


5149 


1-387177 


8-613189^5158 


11-386811 


•000365 


08 


9-999635 


39 


22 


8-615891 


5112 


1-384109 


8-6l6262|5121 


11-383738 


-000371 


10 


9-999629 


38 


23 


8-618937 


5076 


1-381063 


8-61931315085 


11-380687 


-000376 


08 


9-999624:137 | 


24! 


8-621962 


5041 


1-378038 


8-622343'5050 


11-377657 


•000381 


08 


9-999619 


36 


25 i' 8-624965 


5006 


1-375035 


8-625352'5015 


11-374648 


-000386 


08 


9-999614 


35 


26 j 


8-627948 


4972 


1-372052 


8-6283404981 


11-371660 


•000392 


10 


9-999608 


34 


27 


8-630911 


4938 


1-369089 


8-631308 4947 


11-368692 


-000397 


08 


9-999603 


33 


28' 


8-633854 


4904 


1-366146 


8-6342564913 


11-365744 


•000403 


10 


9-999597 


32 


29:' 8-636776 


4871 


1-363224 


8-637184 4880 


11-362816 


-000408 


08 


9-999592 


31 


30^^8-639680 


4839 


1-360320 


8-64009314848 


11-359907 


-000414 


08 


9-999586 


30 


31 |i 8-642563 


4806 


1-357437 


8-642982 


4816 


11-357018 


•000419 


10 


9-999581 


29 


32 


8-645428 


4775 


1-354572 


8-645853 


4784 


11-354147 


-000425 


10 


9-999575 


28 


33 


8-648274 


4743 


1-351726 


8-648704 


4753 


11-351296 


-000430 


08 


9-999570 


27 


34 


8-65110214712 


1-348898 


8-651537 


4722 


11-348463 


-000436 


10 


9-999564 


26 


35 


8-653911 


4682 


1-346089 


8-654352 


4691 


11-34.5648 


-000442 


10 


9-999558 


25 


36 


8-656702 


4652 


1-343298 


8-657149 


4661 


11-342851 


-000447 


08 


9-999553 


24 


37 


18-659475 


4622 


1-340525 


8-659928 


4631 


11-340072 


-000453 


10 


9-^9547 


23 


38 


8-662230 


4592 


1-337770 


8-662689 


4602 


11-337311 


■000459 


10 


9-999541 


22 


39 


8-664968 


4563 


1-335032 


8-665433 


4573 


11-334567 


-000465 


10 


9-999535 


21 


40 


8-66768914535 


1-332311 


8-668160 


4544 


11-331840 


-000471 


10 


9-999529 


20 


41 


8-670393!4506!l-329607 


i 8-670870 


4516 


11-329130 


-000476 


08 


9-999524 


19 


42 


8-673080 


4479 


1-326920 


8-673563 


4488 


11-326437 


-000482 


10 


9-999518 


18 


43 


8-675751 


4451 


1-324249 


1 8-676239 


4461 


11-323761 


-000488 


10 


9-999512 


17 


44118-678405 


4424 


1-321595 


1 8-678900 


4434 


11-321100 


•000494 


10 


9-999506 


16 


45 


8-681043 
B-683665 


4397 


1-318957 


; 8-681544 


4407 


11-318456 


•000500 


10 


9-999500' |15 


46 


4370 


1-316335 


: 8-684172 


4380 


11-315828 


•000507 


10 


9-999493||14 


47 


8-686272 


4344 


1-313728 


8-686784 


4354 


11-313216 


•000513 


12 


9-999487 


13 


48 


8-688863 


4318 


1-311137 


8-689381 


4328 


11-310619 


•000519 


10 


9-999481 


12 


49 


8-691438 


4292 


1 1-308562 


8-691963 


4303 


11-308037 


•000525 


10 


9-999475 


11 


50 


8-693998 


4267 


11-306002 


8-694529 


4277 


11-305471 


-000531 


10 


9-999469 


10 


51 


8-696543 


4242 


11-303457 


8-697081 


4252 


11-302919 


•000537 


12 


9-999463 


9 


52 


8-699073 


4217 


:l-300927 


8-G99617 


4228 


11-300383 


-000544 


10 


9-999456 


8 


53 


8-701589 


4192 


il-298411 


8-702139 


4203 


11-297861 


-000550 


10 


9-999450 


7 


54 


8-704090 


4168 


'1-295910 


8-704646 


4179 


11-295354 


-000557 


10 


9-999443 


6 


55 


8-706577 


4144 


1-293423 


8-707140 


4155 


11-292860 


-000563 


12 


9-999437 


5 


56 


8-709049 


4121 


11-290951 


8-709618 


4132 


11-290382 


-000569 


10 


9-999431 


4 


57 


8-711607 


4097 


1-288493 


8-712083 


4108 


U-287917 


-000576 


12 


9-999424 


3 


58 


8-713952 


4074 


1-286048 


8-714534 


4085 


11-285466 


-00058ii 


10 


9-999418 


2 


59 


8-716383 


4052 


1-283617 


8-716972 


4062 


11-283028 


-000589 


12 


9-999411 


1 


60 


8-718800'4029 


1-281200 8-719396 


4040 


11-280604 


-000596 


12 


9-999404 





. ' 


Cosine. Secant. Colangcnt. | 


Tangent. 


Cosecant. 




Sine. 


31 



87 DEG. 



44 



LOGARITHMIC SINES, ETC. 



3 DEC. 























Sine. 


Diff. 
100" 


Cbsecant. 


Tangent. 


for 1 cotangent. 


Secant. 


Diff. 

100" 


cosine. 


60 


8-718800 




1-281200 


18-719396 


!U -280604 


-000596 




9-999404 


1 


8-721204 


4006 


1-278796! 8-721806 


4017 11-278194 


-000602 


10 


9-999398 


59 


2 


8-723595 


3984 


1-276405 8-724204 


3995111-275796 


-000609 


12 


9-999391 


158 


3 


8-725972 


39621-274028 8-726588 


3974 11-273412 


-000616 


12 


9-999384157 i 


4 


8-728337 


394l!l-271663 8-728959 


395211-271041 


•000622 


10 9-999378156 


5 


8-730688 


3919|1-269312 8-731317 


3931 11-268683 


-000629 


12 9-999371 55 


6 


8-733027 


3898il-266973i 8-733663 


3909:11-266337 


-000636 


12 9-999364|54 


7 


8-735354 


3877 


1-264646] 8-735996 


3889:11-264004 


-000643 


12 9-999357 53 


.8 


8-737667 


3857 


1-262333 8-738317 


3868'11-261683 


-000650 


12 9-999350152 


9 


8-739969 


3836 


1-260031 8-740626 


3848|ll-259374 


-000657 


12 9 -9993431 51 


10 


8-742259 


3816 


1-257741 


i 8-742922 


3827 11-257078 


-000664 


12 9-999336['50 


11 


8-744536 


3796 


1-255464 


1 8-745207 


3807!ll-254793 


-000671 


12 9-999329|J49 
12 9-999322|'48 


12 


8-740802 


3776 


1-253198 ! 8-747479 


3787 11-252521 


-000678 


13 


8-749055 


3756 


1-250945 li 8-749740 


3768!ll-250260 


-000685 


12 9-999315|47 


14 


8-751297 


3737 


1-248703118-751989 
1-246472' 8-754227 


3749!ll-248011 


-000692 


12 9-9993081 46 


15 


8-753528 


3717 


3729 11-245773 


-000699 


12 9-99930l|:45 


16 


8 -705747 


3698 


1-244253' 8-756453 


3710 11-243547 


-000706 


12 9-999294!!44 


17 


8-757955 


367911-242045;! 8-758668 


3692!ll-241332 


-000713 


13 9-999287 


43 


18 


8-760151 


3661 


1-239849 !i 8-760872 


3673 11-239128 


-000721 


12 9-999279! 


42 


19 


8-762337 


3642 


1-237663 '18-763065 


3655111-236935 


-000728 


12 9-999272 


41 


20 


8-764511 


3624 


1-235489 : 8-765246 


3636 


11-234754 


-000735 


12 9-999265 


40 


21 


8-766675 


3606 


1-233325 8-767417 


3618 


11-232583 


-000743 


13 9-999257 


39 


22 


8-768828 


3588 


1-231172. 8-769578 


3600 


11-230422 


-000750 


12 9-999250:38 


23 


8-770970 


3570 


1-229030 


18-771727 


3583 


11-228273 


-000758 


13 9-999242|l37 


24 


8-773101 


3553 


1-226899 


8-773866 


3565111-226134 


-000765 


12 9-999235 |36 


25 


8-775223 


3535 


1-224777 


8-775995 


3548 


11-224005 


-000773 


13 9-999227 


35 


26 


8-777333 


3518 


1-222667 ji 8-778114 


3531 


11-221886 


-000780 


12 9-999220 


34 


27 


8-779434 


3501 


1-220566 


8-780222 


3514 


11-219778 


-000788 


13 9-999212!'33 


28 


8-781524 


3484 


1-218476 


8-782320 


3497ill-217680 


-000795 


13 9-999205' 32 


29 


8-783605 


3467 


1-216395 


8-784408 


3480 ! 11 -215592 


-000803 


13 9-999197 31 


30 


8-785675 


3451 


1-214325 


8-786486 


3464lll-213514 


-000811 


13 9-999189 30 


31 


8-787736 


3434 


1-212264 


8-788554'3447|ll-211446 


-000819 


13 9-999181129 


32 


8-789787 


3418 


1-210213 


8-790613 3431 11-209387 


-000826 


12 |9-999174 i28 


33 


8-791828 


3402 


1-208172 


l8-792662;3415 11-207338 


-000834 


13 19-999166 '27 


34 


8-793859 


3386 


1-206141 


! 8-794701:3399 11-205299 


-000842 


13 


9-999158 26 


35 


8-795881 


3370 


1-204119 


I 8-79673l|3383 11-203269 


-000850 


13 


9-999150:25 


36 


8-797894 


3354 


1-202106 


!8-798752i3368!ll-201248 


-000858 


13 


9-999142 !24 


37 


8-799897 


3339 


1-200103!! 8-80076313352 


11-199237 


-000866 


13 


9-999134 23 


38 


8-801892 


3323 


1-198108 i!8-802765|3337 


11-197235 


-000874 


13 


9-999126 22 


39 


8-803876 


3308 


1-196124118-80475813322 


11-195242 


-000882 


13 


9-999118 '21 


40 


8-805852 


3293 


1-194148 


i8-806742!3306 


11-193258 


-000890 


13 


9-999110 20 


41 


8-807819 


3278 


1-192181 


8-808717 3292 


11-191283 


-000898 


]3 


9-999102 19 


42 


8-809777 


3263 


1-190223 


8-810683 3277 


11-189317 


•000906 


13 


9-999094 18 


43 


8-811726 


3249 


1-188274 


8-81264l|3262 


11-187359 


-000914 


13 


9-999086 17 


44 


8-813667 


3234 


1-186333 


8-814589i3248 


11-185411 


-000923 


15 


9-999077 16 


45 


8-815599 


3219 


1-184401 


8-816529,3233 


11-183471 


•000931 


13 


9-999069 


15 


46 


8-817522 


3205 


1-182478 


8-8184613219 


11-181539 


•000939 


13 


9-99906t 


14 


47 


8-819436 


3191 


1-180564 


8-820384I3205 


11-179616 


•000947 


15 


9-999053 


13 


48 


8-821343 


3177 


1-178657 


8-822298i3191 


11-177702 


•000956 


13 


9-999044 


12 


49 


8-823240 


3163 


1-176760 


8-824205|3177 


11-175795 


•000964 


13 


9-999036 


11 


50 


8-825130 


3149 


1-174870 


8-826103I3163 


11-173897 


•000973 


15 


9-999027 


10 


51 


8-827011 


3135 


1-172989 


8-827992 


3150 


11-172008 


•000981 


13 


9-999019 


9 


52 


8-828884 


3122 


1-171116 


8-829874 


3136 


11-170126 


•000990 


15 


9-999010 


8 


53 


8-830749 


3108 


1-169251 


8-831748 


3123 


11-168252 


•000998 


13 


9-999002 


7 


54 


8-832607 


3095 


1-167393 


18-833613 


3109 


11-166387 


•001007! 15 '9-998993 1 6 


55 


8-834456 


3082 


1-165544 ji 8-835471 


3096 


11-164529 


•001016, 15 9-998984J 5 


56 


8-836297 


3069 


1-1637031 8-837321 


3083 


11-162679; 


-001024; 13 19-998976 I 4 


57 


8-838130 


3056 


1-101870: 8-839163 


3070 


11-160837 i 


-001033 


15 


9-998967 1 3 


58 


8-83995613043 


1-160044! 8-840998 


3057 


11-159002; 


-001042 


15 


9-9989581 2 


59 


8-841774 


3030 


1-158226! 8-842825 30i5 


11-157175! -001050 


13 


9-998950 1 


60 


8-843585 


3017 


1-156415! 8-844644 3032 


11-155356 -001059 


15 J9-998941 1 ! 


/ 


Cosine. 


1 Secant. || Cotangent. 1 | Tangent. ! Cosecant. j Sine. ! ' | 



86 DEC. 



LOGARITHMIC SINES, ETC. 



45 



4 DEG. 






Sine. 


Dift-. 
lOU" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


60 


8-843585 




1-156415 


8-844644 




11-155356 


•001059 




9-998941 


1 


8-845387 


3005 


1-154613 


8-846455 


3019 


11-163645 


-001068 


15 


9-9989321 


59 


2 


8-847183 


2992 


1-152817 


8-848260 


3007 


11-151740 


-001077 


16 


9-9989231 


58 


3 


8-848971 


2980 


1-151029 


8-850057 


2995 


11-149943 


-001086 


15 9^998914 


57 


4 


8-850751 


2967 


1-149249 


8-851846 


2982 


11-148154 


-001095 


16 9-998905 


56 


5 


8-852525 


2955 


1-147475 


8-853628 


2970 


11-146372 


-001104 


16 '9-998896ii55 | 


6 


8-854291 


2943 


1-145709 


8-855403 


2958 


11-144597 


-001113 


15 


9-998887 


54 


7 


8-856049 


2931 


1-143951 


8-857171 


2946 


11-142829 


■001122 


15 


9-998878 


53 


8 


8-857801 


2919 


1-142199 


8-858932 


2935 


11-141068 


-001131 


16 


9-998869 


52 


9 


8-859546 


2908 


1-140454 


8-860686 


2923 


11-139314 


-001140 


16 


9-998860 


51 


10 


8-861283 


2896 


1-138717 


8-862433 


2911 


11-137567 


•001149 


15 


9-998851 


50 


11 


8-863014 


2884 


1-136986 


8-864173 


2900 


11-135827 


-001159 


17 


9-998841 


49 


12 


8-864738 


2873 


1-135262 


8-865906 


2888 


11-134094 


-001168 


15 9-998832] 


48 


13 


8-866455 


2861 


1-133545 


8-867632 


2877 


11-132368 


-001177 


15 


9-998823 


47 


14 


8-868165 


2850 


1-131836 


8-869351 


2866 


11-130649 


-001187 


17 


9-998813 


46 


15 


8-869868 


2839 


1-130132 


8-871064 


2854 


11-128936 


•001196 


15 


9-998804 


45 


16 


8-871565 


2828 


1-128435 


8-872770 


2843 


11-127230 


-001205 


15 


9-998795 


44 


17 


8-873255 


2817 


1-126746 


8-874469 


2832 


11-125631 


-001215 


17 


9-998785 


43 


18 


8-874938 


2806 


1-125062 


8-876162 


2821 


11-123838 


-001224 


15 


9-998776 


42 


19 


8-876615 


2795 


1-123385 


8-877849 


2811 


11-122151 


-001234 


17 


9-998766 


41 


20 


8-878285 


2784 


1-121715 


8-879529 


2800 


11-120471 


-001243 


15 


9-998767 


40 


21 


8-879949 


2773 


1-120051 


8-881202 


2789 


11-118798 


-001253 


17 j9-998747l 


39 


22 


8-881607 


2763 


1-118393 


8-882869 


2779 


11-117131 


-001262 


17 


9-998738] 


38 


23 


8-883258 


2752 


1-116742 


8-884530 


2768 


11-115470 


-001272 


15 


9-998728i 


37 


24 


8-884903 


2742 


1-115097 


8-886185 


2758 


11-113815 


-001282 


17 


9-998718 


36 


25 


8-886542 


2731 


1-113458 


8-887833 


2747 


11-112167 


-001292 


17 


9-998708 


35 


26 


8-888174 


2721 


1-111826 


8-889476|2737 


11-110524 


-001301 


16 


9-998699: 


34 


27 


8-889801 


2711 


1-110199 


8-891112:2727 


11-108888 


-001311 


17 


9-998689 


33 


28 


8-891421 


2700 


1-108579 


8-89274212717 


11-107258 


•001321 


17 


9-998679 


32 


29 


8-893035 


2690 


1-106965 


8-8943662707 


11-105634 


-001332 


17 


9-998669 


31 


30 


8-891643 


2680 


1-105357 


8-89598412697 


11-104016 


•001341 


17 


9-998659 


30 


31 


8-896246 


2670 


1-103754 


8-897596 


2687 


11-102404 


•001361 


17 


9-998649 


29 


32 


8-897842 


2660 


1-102158 


8-899203 


2677 


11-100797 


-001361 


17 


9-998639 


28 


33 


8-899432 


2651 


1-100568 


8-900803 


2667 


11-099197 


•001371 


17 


9-998629 


27 


34 


8-901017 


2641 


1-098983 


8-902398 


2658 


11-097602 


•001381 


17 


9-998619 


26 


35 


8-902596 


2631 


1-097404 


8-903987 


2648 


11-096013 


•001391 


17 


9-998609 


25 


36 


8-904169 


2622 


1-095831 


8-905570 


2638 


11-094430 


•001401 


17 


9-998599 


24 


37 


8-905736 


2612 


1-094264 


8-907147 


2629 


11-092853 


-001411 


17 


9-998589 


23 


38 


8-907297 


2603 


1-092703 


8-908719 


2620 


11-091281 


•001422 


18 


9-998578 


22 


39 


8-908853 


2593 


1-091147 


8-910285 


2610 


11-089715 


-001432 


17 


9-998668 


21 


40 


8-910404 


2584 


1-089596 


18-911846 


2601 


11-088164 


•001442 


17 


9-998558 


20 


41 


8-911949 


2576 1-088051 


! 8-913401 


2592 


11-086599 


•001452 


17 


9-998548 


19 


42 


8-913488 


2566 1-086512 


1 8-914951 


5283 


11-085049 


•001463 


18 


9-998537 


18 


43 


8-915022 


2556 1-084978 


' 8-916495 


2574 


11-083605 


•001473 


17 


9-998627 


17 


44 


8-916550 


2547 1-083450 


8-918034 


2565 


11-081966 


•001484 


18 


9-998516 


16 


45 


8-918073 


2538 1-081927 


8-919668 


2656 


11-080432 


•001494 


17 


9-998506 


15 


46 


8-919591 


252911-080409 


8-921096 


2547 


11-078904 


-001605 


18 


9-998496 


14 


47 


8-921103 


2520 


1-078897 


8-922619 


2538 


11-077381 


•001616 


17 


9-998485 


13 


48 


8-922610 


2512 


1-077390 


8-924136 


2530 


11-075864 


•001526 


18 


9-998474 


12 


49 


8-924112 


2503 


1-075888 


8-925649 


2621 


11-074351 


•001536 


17 


9-998464 


11 


50 


8-925609 


2494 


1-074391 


8-927156 


2512 


11-072844 


-001547 


18 


9-998453 


10 


51 


8-927100 


2486 


1-072900 


8-928658 


2503 


11-071342 


•001558 


18 


9-998442 


9 


52 


8-928587 


2477 


1071413 


8-930155 


2495 


11-069845 


•001569 


18 


9-998431 


8 


53 


8-930068 


2469! 1-069932 


8-931647 


2486 


11-068353 


•001579 


17 


9-998421 


7 


54 


8-931544 


24601 1-068456 


8-933134 


2478 


11-066866 


'•001590 


18 


9-998410 


6 


55 


8-933015 


2452 


! 1-066985 


8-934616 


2470 


11-065384 


•001601 


18 


9-998399 


5 


56 


8-934481 


2443 


1-065519 


8-936093 


2461 


11-063907 


•001612 


18 


9-998388 


4 


57 


8-935942 


12435 


1-064058 8-937565 


2453 


11-062435 


•001623 


18 


9-998377 


3 


58 


8-93739812427 


1-062602 8-939032 


2445 


11-060968 


•001634 


18 


9-998366 


2 


69 


8-938850 


2419 


1-061150 8-940494'2437 


11-059506 


•001646 


18 


9-998356 


1 


60 


8-94029b 


2411 


1-059704 8-94195252429 


11-058048 


•001656 


18 


9-998344 





Cosine. 


1 : Secant. Cotangent. 1 


Tangent. 


Cosecant. 




Sine. 


' 



85 D£a. 



46 



LOGARITHMIC SINES, ETC. 




8i DE6. 



LOGARITHMIC SINES, ETC. 



47 



6 DE6. 



1 , 

i 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


m'' Cotangent. 


! Secant. 


Diff 
100" 


Cosine. 


!■ 

1 


i 


9-019235 




•980765 


9-021620 


'10-978380 


i -002386 




9-997614 


60 


1 


9-020435 


2000 


-979565 


9-022834 


2023 10-977166 


-002399 


22 


9-997601 


|59 


2 


9-021632 


1995 


-978368 


9-024044 


201710-975956 


•002412 


22 


9-997588 


58 


3 


9-022825 


1989 


•977175 


9-025251 '2011 10-974749 


-002426 


23 


9-997574 


57 


4 


9-024016 


1984 
1978 


-975984 


9-0264551200610-973545 


1-002439 


22 


9-997561 


56 


5 


9 •02-5203 


-974797 


9-027655i2001 10-972345 


-002453 23 


9-997547 


55 


•6 


9-026386 


1973 


•973614 


9-028852 


1995*10-971148 


-002466' 22 


9^997534 


54 


7 


9-027567 


1967 


-972433 


9-030046 


199010-969954 


•002480 23 


9^997520 


53 


8 


9-028744 


1962 


-971256 


9^031237 


1985*10-968763 


•002493 


22 


9^997507 


52 


9 


9-029918 


1957 


-970082 


9-032425 


1979 10-967575 


•002507 


23 


9^997493 


51 


10 


9-031089 


1951 


-968911 


9-033609 


1974 10-966391 


•002520 


22 


9-997480 50 


11 


9-032257 


1946 


-967743 


9-0347911969 10-965209 


•002534 


28 


9-997466 49 


12 


9-033421 


1941 


•966579 


9-035969 1964 10-964031 


1-002548 


23 


9-997452 


48 


13 


9-034582 


1936 


-965418 


9-037144 1958 10-962856 


1-002561 


22 


9-997439 


47 


14 


9-035741 


1930 


•964259 


9-038316 1953 10-961684 


•002575 


23 


9-997425 


146 


15 


9-036896 


1925 


•963104 


9-039485|1948l0-960515 


•002589 


23 


9-997411 


45 


16 


9-038048 


1920 


•961952 


9-04065111943 10-959349 


•002603 


23 


9-997397 


'44 


17 


9-039197 


1915 


-960803 


9-0418131938 10-958187 


•002617 


23 


9-997383 


43 


18 


9-040342 


1910 


•959658 


9-0429731933 10-957027 


-002631 


23 


9-997369 


42 


19 


9-041485 


1905 


•958515 


9-0441301928 10-955870 


•002645 


23 


9-997355 


41 


20 


9-042625 


1899 


•957375 


9-045284{1923 10-954716 


-002659 


23 


9-997341 


40 


21 


9-043762 


1895 


•956238 


9-0464341918 10-953566 


-002673 


23 


9-997327 


39 


22 


9-044895 


1889 


•955105 i: 9-0475824913 10-952418 


•002687 


23 


9-997313 38 


23! 


9-046026!l884 


-953974 ! 9-0487271908'l0-951273 


•002701 


23 


9-997299 37 


24 


9-04715411879 


•952846 i 9-049869 1903:10-950131 


•002715 


23 


9-997285 36 


25 


9-048279 1875 


-951721 i 9-0510081898 10-948992 
-950600 !9-052144ll893'l0-947856 


•002729 


23 


9-997271 35 


26 


9-049400 1870 


-002743 


23 


9.997257 34 


27 


9-050ol9[1865 


•949481 1 9^053277 l889'l0-946723 


-002758 


25 


9-997242 33 


28 


9 -051 635! I860 


•948365 i 9^054407 1884!l0-945593 


-002772' 23 


9-997228 32 


29 1 


9-052749 1855 


•947251 


9-0.55535187910-944465 


-002786 23 


9-997214 31 


30 


9-053859 


1850 


•946141 


9-0566501874!l0-943341 


-002801 25 


9-997199||30 


31 


9-054966 


1845 


-945034 


9-057781l870'l0-9422]9 


•002815 23 


9-997185 


29 


32 


9-0-56071 


1841 


•943929 ! 9-058900 1865 10-941100 


-0028301 25 


9-997170 


28 


33 


9-05717211836 


-942828 : 9-060016 1860 10-939984 


-002844 23 


9-997156 


127 


34 


9-05827111831 


•941729 ' 9-061130 1855 10-938870 


-0028591 25 


9-997141 


26 


35 


9-059367 


1827 


•940633 ' 9-062240 1851 10-937760 


•002873; 23 


9-997127 


25 


36 


9-060460 


1822 


•939540 i: 9-0633481846:10-936652 


•002888 25 


9-997112 


24 


37 


9-061551 


1817 


•938449 \\ 9-064453 l842!lO-935547 


-002902 


23 


9-997098 


23 


38 


9-062639 


1813 


•937361 p 9-065556!l837il0-934444 


-002917 


25 


9-997083 


22 


39' 


9-063724 


1808 


-936276 ij 9-0666-55I183310-933345 


•002932 


25 


9-997068 


21 


40 


9-064806il804 


•935194 { 9-0677521828 10-932248 


-002947 


25 '9-997053 


20 


41 


9-065885 


1799 


•934115 9-068846 182410-931154 


-002961 


23 


9-997039 


19 


42 


9-066962 


1794 


-933038 9-069938 


181910-930062 


-002976 


25 


9-997024 


18 


43 


9-068036 


1790 


-931964' 9-071027 


1815110-928973 


-002991 


25 


9-997009 


17 


44 


9-069107 


1786 


-930893 9-072113 


1810 10-927887 


-003006 


25 


9-996994 


16 


45 i 


9-070176 


1781 


-929824 


9-0731971806 10-926803 


-003021 


25 


9-996979 


15 


46 i 


9-071242 


1777 


-928758 


9-074278!l802'lO-925722 


-003036 


25 


9-996964 


14 


47 j 9-072306 


1772 


•927694 


9-075356 1797!l0-024644 


-003051 


25 


9-996949 


U 


48 


9-073366 


1768 


-926634 


9-076432 1793il0-923568 


-003066 


25 


9-996934 


12 


49 


9-074424 


1763 


•925576 


9-0775051789 10-922495 


-003081 


25 


9^996919 


11 


50 


9-075480 


1759 


•924520 


9.078576 1784110-921424 


•003096 


25 


9^996904 


10 


51 


9-076533 


1755 


•923467 


9-079644178010-920356 


•003111 


25 


9^996889 


9 


52 


9-077583 


1750 


-922417 


9-0807101776 10-919290 


•003126 


25 


9^996874 


8 


53 


9-078631 


1746 


•921369 


9-08l7731772'lO-918227 


-003142 


27 


9-996858 


7 


54 


9-079676 


1742 


•920324 


9-082833 1767 


10-917167 


•003157 


25 


9-996843 


6 


55 


9-080719 


1738 


•919281 


9-083891 1763 


10-916109 


•003172 


25 


9^996828 


5 


56 


9-0817.59il733 


•918241 ! 


9-084947 1759 


10 9150-53 


•003188 27 


9-996812 


4 


57 


9 -082797 il72<^ 


•917203 19-08600011755 


10-U14000 


•003203' 25 


9^996797 


3 


58 


9-0832321725 


•916168 'i9-0870o01751 


10-9)2950 


-003218! 25 19-996782 


2 


59 


9-084864 


1721 


•915136 'i9^088(>98|l747 


10-911902 


-0032341 27 


9^996766 


1 


60 


9-085894 


1717 


-914106 l' 9-089144!l74310-9108.56 ';-003249! 25 


9^996751 





' 1 


Cosine. 


Secant. CDtangmt. 'J'angeiit. ' Cosucant. 


Sine. 


' ■■ 



83 DEQ. 



48 



LOGARITHMIC SINES, ETC. 



7 DEC. 



. i 

J 


Sine. 


Diff. 
100" 


Cosecant. , 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


Pol)"i Cosine. 


' 





9-085894 




•914106 1 


9-089144| 


10-910856 


-003249 


|9-99675l|l60 


1 


9-086922 


1713 


•913078 


9-0901871738 


10-909813 


•003265 


27 !9-996735':59 


2 


9-087947 


1709 


•912053 


9-091228 1735 


10-908772 


-003280 


25 9^9967201:58 


3| 


9-088970 


1704 


•911030' 


9-092266 1731 


10-907734 


-003296 


27 


9-996704 57 


41 


9-089990 


1700 


•910010 1 


9-093302 


1727 


10-906698 


-003312 


27 


9-996688.56 


5! 


9-091008 


1696 


•908992 


9-094336 


1722 


10-905664 


-003327 


25 


9-996673 55 


6! 


9-092024 


1692 


•907976 i 


9-095367 


1719 


10-904633 


-003343 


27 


9-996657 54 


7 9-093037 


1688 


-906963 i 


9-096395 


1715 


10-903605 


-003359 


27 


9-996641'53 


8! 9-094047 1684 


•905953 1 


9-097422 


1711 


10-902578 


-003375 


27 


9-996625 :52 


9:!9-095056'l680 


•904944 1 


9-098446 


1707 


10-901554 


-003390 


25 


9-996610151 


10 Jl 9-096062 1676 


•903938 ' 


9-099468 


1703 


10-900532 


-003406 


27 


9-996594 50 


lli|9-097065!l673 


-902935 1 


9-100487 


1699 


10-899513 


-003422 


27 


9-996578 49 


12^19-098066 


1668 


•901934 


9-101504 


1695 


10-898496 ; 


-003438 


27 


9-996562 48 


13;' 9-099065 


1665 


•900935 


9-102519 


1691 


10-897481 ! 


003454 


27 


9-996546 47 


14 ;i 9-100062 


1661 


-899938 


9-103532 


1687 


10-896468 


•003470 


27 


9-996530 ,46 


15 il 9-101056 

16 19-102048 


1657 


•898944 


9-104542 


1684 


10-895458 


-003486 


27 


9-996514145 


1653 


•897952 


9-105550 


1680 


10-894450 


-003502 


27 


9-996498 144 


nil 9-103037 


1649 


•896963 


9^106556 


1676 


10-893444 


■003518 


27 


9-996482 43 


18 '19-104025 


1645 


•895975 


9-107559 


1672 


10-89^441 


-003535 


28 


9-996465 42 


19 i 9-105010 


1642 


•894990 


9-108560 


1669 


10-891440 


-003551 


27 


9-996449 41 


20 il 9-105992 


1638 


•894008 


9-109559 


1665 


10-890441 


-003567 


27 


9-996433/40 


21119-106973 


1634 


•893027 


9-110556 


1661 


10-889444 


•003583 


27 


9-99641739 


22 1 9-107951 


1630 


•892049 


9-111551 


1658 


10-888449 


-003600 


28 


9-996400 38 


23119-108927 


1627 


•891073 


9-112543 


1654 


10-887457 


•003616 


27 


9-996384' 37 


24 Ij 9-109901 


1623 


•890099 


9-113533 


1650 


10-886467 


•003632 


27 


9-996368 36 


251 


9-110873 


1619 


•889127 


9-114521 


1647 


10-885479 : 


-003649 


28 


9-9963511135 


26 1 


9-111842 


1616 


•888158 


9-115507 


1643 


10-884493 


•003665 


27 


9-996335; 34 


27 1 


9-112809 


1612 


•887191 


9-116491 


1639 


10-883509 


•003682, 28 


9-996318133 


28 


9-113774 


1608 


•886226 


9-117472 


1636 


10-882528 


-003698' 27 


9-996302 32 


29! 


9-114737 


1605 


•885263 


9-118452 


1632 


10-881548 


-003715 28 


9-996285 31 


30 1 


9-115698 


1601 


-884302 


9-119429 


1629 


10-880571 


-0037311 27 


9-996269:30 


311 


9-116656|l597 


-883344 


9-120404 


1625 


10-879596 


-003748' 28 


9-996252 29 


32i!9-117613jl594| -882387 


9-121377 


1622'l0-878623l 


-0037651 28 


9-996235 28 


33 |; 9-118567 15901 -881433 


9-1223481161810-8776521 


•0037811 27 


9-996219 27 


34 j 9-119519 1587 


-880481 


9-123317 


1615 


10-8766831 


•003798 


28 


9-996202 26 


35! 9-120469 1583 


-879531 


9-124284 


1611 


10-8757161 


•003815 


28 '9^996185,i25 


361 9-121417 1580 


-878583 


9-125249 


1608 10-874751 


-003832 


28 |9^996168 !24 


37 9-122362 1576 


-8776381 


9-126211 


1604 10-873789 


•003849 


28 9^996151 123 


38119-123306 1573 


-8766941 


9-1271721160110-872828 


-003866 


28 9-996134122 


391,9-12424815691 -875752 


9-128130il597 10-871870 


•003883 


28 9-996117 121 


40 


9-125187 15661 -8748131 


9-129087 1594 10-870913 


•003900 


28 9-996100120 


41 


9-126125 15621 -873875 


9-13004lll591 10-869959 


-003917 


28 9-990083'll9 


42 


9-127060:15591 -8729401 


9-130994ll587 10-869006 


-003934' 28 9^996066ll8 


43 


9-127993 15561 -872007 I 


9-131944.1584 10-868056 


•003951 28 '9-996049 117 


44 


19-128925 1552! -871075 ^ 


9-132893 1581 10-867107 


•003968 28 9-996032 16 


45!! 9-129854 1549i -870146 


9-133839 1577 10-866161 


-003985 28:9-996015|l5 


46 ; 9-130781 1545; -869219 


9-134784,1574 10-865216 


•004002 28 9^995998;ll4 


47 


19-131706 1542: -868294, 


9-135726 


1571jl0-864274 


•004020| 30 9^995980: 13 


48 


1 9-1326301539: -867370 


9-136667 


156710-863333 


•0040371 28 9^995963 12 


49 


19-13355115351 -866449 


9^137605 


1564:10-862395 


•004054; 28 i9-995946||ll 
-004072 30 9-995928110 


50 


1 9-134470 1532 -865530 


9-138542 


1561|10-861458 


51 11 9-135387 15291 -864613 


9-139476 


155810-860524 


-004089 28 19-995911 
•004106 28 19-995894 


9 


52 


19-136303 15251 -863697 


9-140409|l55510-859591 


8 


53 


9-137216 1522; -862784 


9-141340 155lllO-858660 


-0041241 30 19-995876 


7 


54 


9-1381281519| -861872 


9-142269 1548|]0-857731 


•004141128 9-995859 


6 


55 


9-139037 1516 -860963 


9-14319611545 10-856804 


-004159 30 9-995841 


5 


56 


9-139944 1512 -860056 


9-144121 1542 10-855879 


-004177 30 9-995823 


4 


57 


9-1408501509 -859150 


9-145044'l539 10-854956 11-004194; 28 19-995806;; 3 | 


58 


9-141754 1506 -858246 


9-14596611535 10-854034 


•004212 30 19^995788, 2 


59 


9-142655 1503 -857345 


9-14688511532 10-853115 


•004229 28|9-99577li 1 


60119-143555 1500 -856445 


i 9-147803,1529 10-852197 |-004247 30 9-995753 





' jl Cosine. Secant. 


Cotangent. ' Tangent. Cosecant. : ' Sine. 


' 



82 DEG. 



LOGARITHMIC SINES, ETC. 



49 



/ 


Sine. 


100" 


Cosecant. 

•856445 


, Tangent, i J^jjJ- ; Cotangent, j 


Secai.t. 


|i|!; Cosine. 


1 , 





9-143555 


|9^147803 


110-852197 j 


•004247 


9 •91)5753 160 


1 


9 


144453 


1496 


-855547 


i9^148718 


1526110-851282] 


-004265 


30 9-9'J57;;5i 59 


2 


9 


145349 


1493 


-854651 


9-149632 


152340-8503681 


-U04283 


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3 


9 


146243 


1490 


•853757!; 9-150544 


1520 10-849456 


-004801 


30 9-995699 57 


4 


9 


147136 


1487 


•852864 |i 9-151454 


1517 10-848546 


-004319 


30 9-995681 56 


6 


9 


148026 


1484 


-851974 9-152363 


1514 10-847637 


-004336 


28 9-995664 155 


6 


9 


148915 


1481 


-8510851,9-153269 


1511110-846731 


-004354 


30 9-995646:54 


7 


9 


149802 


1478 


-850198 || 9-154174 


1508il0-845826 


•004372 


30 9-995628,53 j 


8 


9 


150686 


1475 


•849314' 9-155077 


1505:10-844923 


-004390 


30 9-995610 


52 


9 


9 


151569 


1472 


•848431 j 9^155978 


1502110-844022 


•004409 


32 9-995591 


51 


10 


9 


152451 


1469 


•847549 ' 9^156877 


1499 10-843123 


-004427 


30 9-995573 50 


Hi 


9 


153330 


1466 


•846670 9-157775 


1496 10-842225 


•004445 


30 9-995555149 


12 i 


9 


154208 


1462 


-845792 1 9-158671 


1493 10-841329 


•004463 


30 9-995537 148 


13 


9 


155083 


1460 


•844917 1 9-159565 


1490 10-840435 


•004481 


30 ,9-995519 


47 1 


14 1 


9 


155957 


1457 


•844043 i 9^160457 


1487 10-839543 


•004499 


30 9-995501 


46 


15 


9 


156830 


1454 


•843170 9-161347 


1484 10-838653 


•004518 


32 9^995482 


45 


16 


9 


157700 


1451 


•8423001 9^162236 


1481110-837764 


•004536 


30 9^995464 


44 


17 


9 


158569 


1448 


•841431 : 9^163123 


1478J10-836877 


■004554 


30 9^995446 


43 


18 


9 


159435 


1445 


•840565 !|9^164008 


1475I1O-835992 


-004573 


32 9-995427 


42 


19 


9 


160301 


1442 


•839699!' 9^164892 


1 473 1 10-835108 


-004591 


30 9-995409 


41 


20; 


9 


161164 


1439 


•838836^ 9-165774 


1470'l0-834226 


-004610 


32 9-995390 


40 


21 


9 


162025 


1436 


•8379751 9-166654 


1467110-833346 


-004628 


30 9-995372 


39 


22 


9 


162885 


1433 


•837115; 9-167532 


1464 10-832468 


-004647 


32 9-995353 


38 


23 


9 


163743 


1430 


•836257 i 9^168409 


1461 10^831591 


•004666 


32 9^995334 


37 


24 


9 


164600 


1427 


•835400 19-169284 


1458 10^830710 


-004684 


30 9-995316 


36 


25 1 


9 


165454 


1424 


-834546 1 9-170157 


1455 10^829843 


-004703 


32 9^995297 


35 


26 1 


9 


166307 


1422 


•8336931 9-171029 


1453 10^828971 


-004722 


32 9^995278 


34 


27 


9 


167159 


1419 


•832841 i 9^171899 


1450il0-828101 


•004740 


30 9-995260 


33 


281 


9 


168008 


1416 


•831992 i: 9-172767 


1447110-827233 


•004759 


32 9-995241 


32 


291 


9 


168856 


1413 


-831144 i 9-173634 


1444 10-826366 


•004778 


32 ,9-995222 


,31 


30 


9 


169702 


1410 


-830298119-174499 


1442 10-825501 


•004797 


32 9^995203 


30 


31 


9 


170547 


1407 


-829453 Jl 9-175362 


1439!l0^824638 


•004816 32 9-995184 


i29 


32 


9 


171389 


1405 


•828611 


19-176224 


143640-823776 


•004835 


32 


9^995165 


28 


33 


9 


172230 


1402 


-827770 


19-177084 


1433 10-822916 


•004854 


32 


9^995146 


27 


34 


9 


173070 


1399 


-826930 


1 9-177942 


1431 10-822058 


•004873 


32 


9^995127 


26 


35 


9 


173908 


1396 


-8260921' 9-178799 


1428 10-821201 


•004892 


32 


9^995108 25 


36 


9 


174744 


1.394 


•8252561' 9-179655 


1425 10-820345 


•004911 


32 


9-995089124 


37 


9 


175578 


1391 


•824422 jl 9^180508 


1423 10-819492 


•004930 32 


9-995070 


23 


38 


9 


176411 


1388 


•823589!' 9-181360 


1420 10-818640 


•004949 32 


9-995051 


22 


39 


9 


177242 


1385 


•82275811 9^182211 


1417 10-817789 


-004968 32 


9-995032 


21 


40 


9 


178072 


1383 


•821928 H 9-183059 


1415 10-816941 


•0049871 32 


9-995013 


20 


41 


9 


178900 


1380 


•821100 1, 9-183907|l412llO-816093 


•005007 33 


9-994993119 


42' 


9 


179726 


1377 


•820274 |l 9^184752ll409 10-815248 


•0050261 32 


9-994974:18 


43 


9 


180551 


1374 


•819449 1 9^185597jl407il0-814403 


•0050451 32 


9-994955 17 


44 


9 


181374 


1372 


•818626 9-186439 1404 10^813561 


-005065 33 


9-994935 16 


45 


9 


182196 


1369 


•817804! 9^187280il402: 10^812720 


•005084 32 


9^994916;il5 


46 


9 


183016 


1367 


•816984 


1 9-188120 1399|10-811880 


•005104 33 


9-9948961114 


47 


9 


183834 


1364 


•816166 


; 9-188958 139610-811042 


•005123 32 


9^994877l!l3 


48 


9 


184651 


1361 


•815349 


19-18979411394110-810206 


•005143! 33 


9^994857!l2 


49 


9 


185466 


1359 


•814534 


19-190629:1391 


10-809371 


•005162 32 


9-994838111 


50 


9 


186280 


1356 


•813720 ji9^19l462ll389 


10-808538 


•005182; 33 


9-994818'10 


51 


9 


187092 


1353 


•812908 9^192294:1386 


10-807706 


•005202! 33 


9-9947981 9 


52 


9 


187903 


1351 


•812097 j9^193124il384 


10-806876 


•005221 


32 


9-9947791 8 
9-994759'| 7 


63 


9 


188712 


1348 


•811288 ,9^193953il381 


10-806047 


•005241 


33 


54 


9 


189519 


1346 


•810481 1 9-1947804379 


10-805220 


•005261 


33 


9-9947391 6 


55 


9 


190325 


1343 


•809675! 9-195606 1376 10-804394 


•005281 


32 


9-9947201 5 


56 


9 


191130 


1341 


•808870 il 9-196430,1374 10-803570 


•005300! 33 


9-994700] 4 


57 


9 


191933 


1338 


-808067 '9-197253 1371 10-802747 \ 


•0053201 33 


9-994680li 3 


58 


9 


192734 


1336 


-807266 ! 9-198074 1369 10^801926l 


•005340, 33 


9-9946601 2 


59 


9 


193534 


1333 


•806466;! 9-198894 1366. 10^801106 


•005360 33 


9-994640 1 


60 


9-194332 


1330 


•805668 19-199713 1364 10-800287 •005380 33 


9-994620' 


1 


C.;siiie 




S.Haut. ! CVtangeiit. ' ' Tangent. Cosecant. ' i tiiuo. 



81 UUG. 



50 



LOaARITHMIC SINES, ETC. 



' 


Siae. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


■ Secant. 


Diff. 
100" 


'' 1 

Cosine, j^ ' j 





9-194332 




•805668 


9^199713 


110-800287 


-005880 




9-994620160 


1 


9-195129 


1328 


•804871 


9-200529 136lil0-799471 


-005400 


33 


9-994600 59 


2 


9-195925 


1326 


•804075 


9-201345 1359 10-798655 


1-005420; 33 


9-994580 58 j 


3 


9-196719 


1328 


•808281 


9-202159I 1356 10-797841 


1-005440 


33 


9-994560 57 


4 


9-197511 


1321 


•802489 


9-20297lll354:10-797029 


1-005460 


83 


9-994540 56 j 


5 


9-198302 


1318 


•801698 


9-203782 135210-796218 


-005481 


35 


9-994519:55 


6 


9-199091 


1316 


•800909 


9-204592| 13491 10-795408 


-005501 


38 


9-994499 54 i 


7 


9-199879 


1313 


•800121 


9-205400 


1347 


10-794600 


1-005521 


33 


9-994479 53 


8 


9-200666 


1311 


•799334 


9-206207 


1845 


10-793793 


i -005541 


33 


9-994459 52 


9 


9-201451 


1308 


•798549 


9-207013 


1342 


10-792987 


-005562 


35 


9-994488 51 


10 


9-202284 


1306 


•797766 


9^207817 


1840 10-792183 


t -0055821 83 


9-994418 50 


11 


9-208017 


1304 


•796983 


9-208619 


1838:10-791381 


i -0056021 33 


9-994398 49 


12 


9-203797 


1301 


•796203 


9-209420 


1885110-790580 


1-005623' 35 


9-994377 48 


• 13 


9-204577 


1299 


•795423 


9-210220 


1333 


10-789780 


! -005648 183 


9-994357 47 


14 


9-205354 


1296 


•794646 


9-211018 


1381 


10-788982 


-005664 


35 


9-994886146 


15 


9-206181 


1294 


•798869 


9-211815 


1328 


10-788185 


1-005684 


88 


9-994816 '45 


16 


9-206906 


1292 


•798094 


9-212611 


1326 


10-787389 


•005705 


35 


9-994295 44 


17 


9-207679 


1289 


•792321 


9-213405 


1324 


10-786595 


-005726 


35 


9-994274143 j 


18 


9-208452 


1287 


•791548 


9-214198 


1321 


10-785802 


•005746 


33 


9-994254142 1 


19 


9-209222 


1285 


•790778 


9-214989 


1319 


10^785011 


•005767 


35 


9-994233141 


20 


9-209992 


1282 


•790008 


9-215780 


1317 10-784220 


•005788 


35 


9-994212 40 


21 


9-210760 


1280 


•789240 


9-216568 


1815 10-783432 


•005809 


35 


9-994191 39 


22 


9-211526 


1278 


•788474 


9-217356 


1312 10-782644 


•005829 


33 


9-994171 38 


23 


9-212291 


1275 


•787709 


9-218142 


1310!10-781858 


1-005850 


35 


9-994l50'87 


24 


9-213055 


1273 


•786945 


9-218926 


1308 10-781074 


-005871 


35 


9-994129 36 


25 


9-213818 


1271 


•786182 


9-219710 


1305 10-780290 


•005892 


35 


9-994108 85 


26 


9-214579 


1268 


•785421 ji 9-220492 


1303 10-779508 


-005913 


35 


9-994087 34 


27 


9-215388 


1266 


•784662 19-221272 


1301 10-778728 


1-005934 


35 


9-994066 '33 


28 


9-216097 


1264 


•783908 


9-222052 


1299!l0-777948 


1-005955 


35 


9-994045:32 


29 


9-216854 


1261 


•788146 


9-222830 


1297!lO-777170 


i-005976 


35 


9-994024 '31 


30 


9-217609 


1259 


•782891 


9-223607 


1294il0-776393 


•005997 


35 


9-994003130 


31 


9-2183631257 


•781637 


9^2-24382 


1292|10-775618 


ft)06018! 35 


9-993982129 


32 


9-219116 


1255 


•780884 


9-225156 


1290' 10-774844 


-006040 37 


9-993960 28 


33 


9-219868 


1253 


•780132 


9^225929 


128810-774071 


-006061 


35 


9-99393927 


34 


9-220618 


1250 


•779382 


9-226700 


1286 10-773300 


•006082 


35 


9-993918 126 


35 


9-221367 


1248 


•778633 


9^227471 


1284!l0-772529 


-006103 


35 


9-993897125 1 


36 


9-222115 


1246 


•777885 


9^228239 


1281il0-771761 


-006125 


37 


9-9938751-24 


37 


9-222861 


1244 


•777139 


9-229007 


1279|10-770993 


•006146 


35 


9-993854123 


38 


9-223606 


1242 


•776894 


9-229773 


1277110-770227 


N006168 


37 


9-993832122 


39 


9-224349 


1239 


•775651 


9-230589 


1275 10-769461 


-006189 


85 


9-998811 121 


40 


9-225092 


1287 


•774908 


9-231802 


1273il0-768698 


•006211 


37 


9-9937891-20 1 


41 


9-225833 


1235 


•774167 


9-232065 


1271 


10-767935 


•006232 


35 


9-993768 19 


42 


9-226573 


1283 


•778427 


9-282826 


1269 


10-767174 


•006254 


37 


9-993746 18 ' 


43 


9-227311 


1231 


•772689 


9-238586 


1267 


10-766414 


•006275 


35 


9-993725,17 1 


44 


9-228048 


1228 


•771952 


9-284345 


1265 


10-765655 


•006297 


37 


9-993703 16 1 


45 


9-228784 


1226 


•771216 


9-285103 


1262 


10-764897 


•006319 


37 


9-993681 15 \ 


46 


9-229518 


1224 


•770482 


9 •235859 


1260 


10-764141 


-006340 


35 


9-993660 14 ! 


47 


9-280252 


1222 


•769748 


9-236614 


1258 


10-763386 


-006862 


37 


9-993638,13 i 


48 


19-230984 


1220 


•769016 


9^287868 


1256 


10-762632 


•006884 


37 


9-993616 12 1 


49 


i 9-231715 


1218 


•768285 ' 


9-238120 


1254 


10-761880 


-006406 


37 


9-993594111 1 


50 


19-232444 


1216 


•767556 


9^238872 


1252 


10-761128 


-006428 


37 


9-993572 10 ' 


51 


! 9-283172 


1214 


•766828 


9-239622 


1250 


10-760378 


•006450 


37 


9-993550 9 | 


52 


;9'288899 


1212 


•766101 


9-240371 


1248 


10-759629 


-006472 


37 


9-993528 8 j 


53 


: 9-234625 


1209 


•765875 


9-241118 


1246 


10-758882 


-006494 87 


9-993506 7 ; 


54 


9-235349 


1207 


•764651 


9-241865 


1244 


10-758135 


■0065161 37 


9-993484 6 ' 


55 


19-286073 


1205 


•763927 


9-242610 


1242 


10-757390 


-0065881 87 


9-993462. 5 ; 


1 56 


1 9-236795 


1203 


•763205 ij 9^243854 


1240 


10-756646 


-006560 37 


9-993440' 4 


' 57 


19-287515 


1201 


•762485 9-244097 


1288 


10-755903 


•006582 37 


9-993418^ 3 1 


58 


; 9-238235 


1199 


•761765 h 9-244839 


1236 


10^755161 


-006604 37 


9-993396: 2 


59 


9-238958 


1197 


•761047 ! 9^245579 


1234 


10-754421 


•0066261 37 


9-993374:! 1 | 


60 


9-289670 


1195 


•760330 i 9^246319 


1282 


10-753681 


•006649J 38 


9-99335li 


i ' 1 Cosine. 


1 Secant. Cotangent, i i Tangent. ! 


Cosecant. | 


Sine. 



80 DE6. 



LOGARITHMIC SIXES, ETC. 



51 



10 


PEG. 




















' 


Sine. 


Dirt. 

m" ■ 


Cosecant. 


Taiigent. 


l),rt. 
luu" 


Cotangent. 


Secant. 


|>i«; cosine. 1' 





9-239670 


i 


•760330 


1 9^246319 




1 0^753681 


•006649 


9-993351 60 | 


1 


9-240386 


1193' 


•759614 


9^247057 


1230 


10^752943 


•006671 


37 9-993329j 


59 


2 


9-241101 


11911 


•758899 


9^247794 


1228 


10^752206 


•006693 


37 9-993307 


58 


3 


9-241814J1189 


•758181 


1 9^248530 


1226 


10^751470 


•006715 


38 9-993285 


57 


4 


9-242526 11871 


•757474 


9^249264 


1224 


10^750736 


•006738 


37 9-993262' 


56 


5 


9-243237 


1185! 


•756763 


1 9^249998 


1222 


10^750002 


•006760 


37 9-993240 


55 j 


6 


9-243947 


1183i 


•756053 


i 9^250730 


1220 


10-749270 


•006783 


38 9^993217 


54 I 


7 


9-244656 


1181! 


•755344 


:9^251461 


1218 


10-748539 


•006805 


37 ,9-993195 


53 1 


8 


9 -245363 ill 79 


•754637 


1 9^252191 


1217 


10^747809 


•006828 


38i9^993172i 


52 


9 


9-24606911177 


•7-53931 


1 9^252920 


1215 


10^747080 


•006851 


38:9^993149 


51 


10 


9-24677511175' 


•753225 


9 •253648 


1213 


10^746352 


•006873 


37|9^993127i 


50 


11 


9-247478:1173 


•752522 


9^254374|1211 


10^745626 


•006896 


38 9^993104 


49 


12 


9-248l8l:117i 


•751819 


9^255100 


1209 


10^744900 


•006919 


38i9-99308li 


4S 


13 


9-248883!ll69 


•751117 


9 •25-5824 


1207 


10^744176 


•006941 


37 9^993059i 


47 


14 


9-24958311167 


•750417 


9 •256547 


1205 


10^743453 


•006964 


38 19-993036; 


46 


15 


9-250282 1165 


•749718 


9^257269 


1203 


10^742731 


-006987 


38i9^9930l3 


45 j 


16 


9 -250980! 1163 


•749020 


9-257990;i201 


10^742010 


•007010 


38 9^992990 


44 1 


171 


9-25167711161 


•748323 


9^258710!1200 


10^741290 


•007033 


38 |9^992967! 


43 1 


18^ 


9-252373 


1159 


•747627 


9-259429ill98 


10^740571 


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38 9^992944| 


42 


19, 


9-253067 


1158 


•746933 


J9-260146 1196 


10-739854 


•007079 


38 9^99292l!;41 1 


20 1 


9-253761 


1156 


•746239 


9-260863;il94 


10-739137 


•007102 


38l9^992898i40 ' 


21 


9-254453 


1154 


•745547 


9-2615781192 


10^738422 


•007125 


38 ;9^992875';39 [ 


22 


9-255144'll52 


•744856 


9-262292 1190 


10-737708 


•007148 


38 19-992852 '38 


23 


9-25583411150 


•744166 


9-263005:1189 


10^736995 


•007171 


38 9-992829 '37 


24 1 


9-2565231148 


•743477 


J9^2637171187 


10-736283 


•007194 


38 ; 9 -992806 36 


25 


9-257211ill46 


•742789 


9^2644281185 


10^735572 


•007217 


38 !9-992783 35 1 


26 


9-257898J1144! 


•742102 


9^265138 1183 


10^734862 


•007241 


40 9^9927-59 34 


27 


9-25858311142 


•741417 


9^265847.1181 


10^7341-53 


•007264 


38 !9^992736 33 


28 


9-259268 1141 


•740732 


9 •266-555 1179 


10^733445 


•007287 


38:9^992713 32 


29i 


9-259951 1139 


•740049 


9^267261ill78 


10^732739 


•007311 


38 |9^992690 31 


30 i 


9-260633 1137 


•739367 


l9-267967ill76 


10^732033 


•007334 


40 


9-992066 30 | 


31 


9-261314 


1135 


•738686 


i9^268671|1174 


10^731329 


•007357 


38 


9^992643 


29 


32 


9-261994 


1133 


•738006 


9^269375jll72 


10-730625 


•007.381 


40 


9^992619 


28 


33 


9-262673 


1131 


•737327 


9^270077I1170 


10-729923 


•007404 


38 


9^992596 


27 


34 


9-263351 


1130 


•736649 


9-2707791169 


10-729221 


h007428 


40 


9-992572J 


26 


35 


9-264027 


1128 


•735973 


9-2714791167 


10-728521 


•007451 


38 


9-992549' 


25 


36 


9-264703 


1126 


•735297 


9-27217811165 


10-727822 


•007475 


40 


9^99252-5| 


24 


37 


9-265377 


1124 


•734623 •9-272876 1164 


10-727124 


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40 


9^992501 


23 


38 


9-266051 


1122 


•733949 9-273573!1162 


10-726427 


•007522 


38 i9-992478 


22 


39 


9-266723 


1120 


•733277 9^274269 


ill60 


10-725731 


•007546 


40 


9-992454 


'21 


40 


9-267395 


1119 


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ill58}10^725036 


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40 


9-992430 


j20 


41 


9-268065 


1117 


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1157 


10-724342 


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40 


9-992406 


119 


42 


9-268734 


1115 


•731266 9^276351 


1155 


10-723649 


•0076181 40 


9-992382 


'l8 


43 


9-269402 


1113 


•730598 9^277043 


1153 


10-722957 


•0076421 38 


9-992358 


17 


44 


9.270069 


nil 


•729931 9-277734ill51 


10^722266 


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40 


9-992335 


16 


45 


9-270735 


1110 


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40 


9-992311 


'15 


46 


9-271400 


1108 


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•007713 


40 


9-992287 


!l4 


47 


9-272064 


1106 


•727936 9-27980l!ll46!l0^720199 


•007737 


40 


9-992263 


!l3 


48 


9-272726 


1105 


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, 9^280488!ll45 10^719512 


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40 


9-992239 


12 


49 


9-273388 


1103 


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10^718826 


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42 


9-992214 


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60 


9-274049 


1101 


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10^718142 


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40 


9-992190 


10! 


51 


9-274708 


1099 


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10^717458 


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9-992166 


9| 


52 


9-275367 


1098 


•724633 9-283225 1138 


10^716775 


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40 


9-992142 


81 


53 


9-276025 


1096 


•723975 9-283907|ll36 


10-716093 


1^007882 


40 


9-992118 


7! 


54 


9-276681 


1094 


•723319 9 •284-388 1135 


10^715412 


N007907 42 


9-992093 


6 


55 


9-277337 


1092 


•722663 9^28-5268 1133 


10^714732 


[•007 931 40 19 -992069 


6 


56 


9-277991 


1001 


•722009 9^2859471131 


10^714053 


1-007956! 42 19-992044 


4 


57 


9-278645 


1089 


•721355 9^286624ill30 


10^713376 


•0079801 40 9-992020 


3 


58 


9-279297 


1087 


•720703 9^287301 11281 10^712699 


•008004; 40 9-991996 


2 


59 


9-279948 


1086 


•7200-32 9-287977I1126 


10^7 12023 i|^008029 


42|9-991971 


1 


60 


9-280599 


1084 


•719101 9-2886.5211125 


10-711348 


•008053 


40|9^991947 





Cosine. 


.Scant. Cjtaii,'ant. t 


Tangent. 


Cosecant.] i Sine. 


"^ 



35 



79 D£6. 



52 



LOGARITHMIC SINES, ETC. 



11 


DEG. 


















' i 


Sine. 


Diff. 
100" 


Cosecant, j | Tangent. 


Difif. 
100" 


Cotangent. I 


Secant. \^^; Cosine. ' i 


o| 


9-280599 




•7194U1 9-288652 




10-7113481 


-008053! 9-991947!i60 


1 


9-281248 


1082 


-718752 9-289326 


1123 


10-710674 :!-008078| 42 i9-991922 59 | 


2 


9-281897 


1081 


•718103' 9-289999 


1122 


10-710001 !!-008103i 42 '9-991897 58 | 


3 


9-282544 


1079 


-717456 9-290671 


1120 


10-709329 N008127i 40 9-991873 57 | 


4 


9-283190 


1077 


-716810 9-291342 


1118 


10-708658 


-0081521 42 19-991848 56 


5 


9-283836 


1076 


•716164 9-292013 


1117 


10-707987! 


-008177; 42 i9-991823:|55 


6 


9-284480 


1074 


-715520 9-292682 


1115 


10-707318! 


-008201 40 9-991799 54 


7 


9-285124 


1072 


•714876 ; 9-293350 


1114 


10-706650 


-008226! 42 9-991774 i53 i 


8 


9-285766 


1071 


•714234 ! 9-294017 


1112 


10-705983 : -0082511 42 9-991749:152 i 


9 


9-286408 


1069 


•713592 i 9-294684 


1111 


10-705316 l!-008276! 42 9-991724-:51 | 


10 


9-287048 


1067 


•712952 i 9-295349 


1109 


10-704651 


•008301142 9-991699 '50 


11 


9-287688il066 


•712312 19-296013 


1107 


10-703987 


•008326; 42 9-991674 49 


12 


9-288326!l064 


-711674 ; 9-296677 


1106 


10-7033-23 


-008351' 42 9-991649 '48 


13 


9-288964 


1063 


-711036 1 9-297339 


1104 


10-702661 


-0083761 42 9-99162447 


U 


9-289600 


1061 


-710400 9-298001 


1103 


10-701999 ' -008401 


42 9-991599 46 


16 


9-290236 


1059 


-709764;! 9-298662 


1101 


10-701338 1-008426 


42 9-991574' 45 


16 


9-290870 


1058 


-709130 1 9-299322 


1100 


10-700678 1-008451 


42 :9-991549;;44 


17 j 9-291504 


1056 


-708496 i! 9-299980 


1098 


10-700020 


-008476 


42 9-991524' 43 


18 i! 9-292137 


1054 


•707863 119-300638 


1096 


10-699362; 


-008502 


43 19-991498 42 


19 


9-292768 


1053 


-707232 19-301295 


1095 


10-698705 ! 


-008527 


42 19-991473 141 


20 


9-293399 


1051 


-706601 19-301951 


1093 


10-698049 ' 


•0085521 42 19-991448 40 | 


21 


9-294029 


1050 


-705971 19-302607 


1092 


10-697393 


-008578 


43 9-991422 39 


221 


9-294658 


1048 


•705342 9-303261 


1090 


10-696739 


-008603 


42 9-991397: 38 


23 j 


9-295286 


1046 


•704714 :|9^303914 


1089 


10-696086 i 


•008028 


42 9-991372!|37 


24! 


9-295913 


1045 


•704087 9^304567 


1087 


10-695433 ; 


•008654 


43 9-991346!|36 


251 


9-296539 


1043 


•703461 '9^305218 


1086 


10-6947821 


-008679 


42 19-991321 


35 


26 


9-297164 


1042 


•702836 9-305869 


1084 


10-694131 


•008705 


43 9-991295 


l34 


27 1 9-297788 


1040 


•702212 9-306519 


1083 


10-693481; 


-008730 


42 9-991270 


133 


28 |9-298412!l089 


-701588 1 9-307168 


1081 


10-692832! 


-008756 


43 9-991244 


32 


29 


9-29903411037 


•700966 9-307815 


1080 


10-692185 


-008782 


43 9-991218:31 i 


30 


; 9-299655! 1036 


•700345 \ 9-308468 


1078 


10-691537 


-008807 


42 9-991193 30 


31 


9-300276 1034 


•6997241 9-309109 


1077 


10-690891 \ 


-008833 


43 9-991167 129 | 


32 


9-300895 1032 


•699105 9-309754 


1075 10-690246^ 


-008859 


43 9-991141 128 | 


33 


9-301514'l031 


-698486] 9-310398 


1074 10-689602 \ 


■008885 


43 9-991115 '27 


34 


9-302132 1029 


-697868; :9-311042 


1073 10-688958 


•008910 


42 9-991090 !26 


35 


9-302748 1028 


•697252 


19-311685 


107110-688315! 


-008936 


43 9-991064 25 ! 


36 


9-303364 1026 


•696636 


'9-312327 


1070 10-687673 [ 


-008962 


43 9-991038; 24 } 


37 


9-303979 1025 


•696021 


'9-312967 


1068 10-687033: 


•008988 


43 9-991012' 23 


38 19-304593 1023 


•695407 


■ 9-313608 


1067 10-686392: 


-009014 


43 9-990986 22 1 


39 19-305207 1022 


•094793 


: 9-314247 


1065 10-685753 


-009040 


43 :9-990960 21 


40 


1 9-305819 1020 


•694181 


9^314885 


1064 


10-685115 ; 


-009066 


43 19-990934:20 


41 


19-306430 1019 


•693570 


9^315523 


1062 


10-684477 ! 


-009092 


43 |9-990908:;19 


42 


9-307041 1017 


•692959 


9-316159 


1061 


10-683841 1 


-009118 


43 19-990882 18 


43 


9-307650 1016 


•692350 


19-316795 


1060 


10-683205 i 


-009145 


45 i9 -990855 17 j 


44 


9-308259 1014 


, ^691741 


9-317430 


1058 


10-682570 


-009171 


43;9-990829 16 


45 


9-308867 1013 


' •691133 


9-318064 


1057 


10-6819361 


-009197 


43 19-990803 15 ' 


46 


9-3094741011 


\ ^690526 


1 9-318697 


1055 


10-681303 


-009223 


43 9-990777|jl4 ' 
45 9-990750 13 1 


47 


9-310080 1010 


•689920 


9-319329 


1054 


10-680671 


-009250 


48 


9-310685 1008 


' -689315 


9-319961 


1053 


10-680039 i 


-009276 


43 '9-99072412 1 


49 


9-311289 1007 


•688711 


1 9-320592 


1051 


10-679408 


•009303 


45 9-990697 


11 ; 


50 


9-311893 1006 


•688107 


j 9-321222 


1050 


10-678778 


-009329 


43 9-990671 


110 ' 


51 


9-312495 1004 


•687505 


9-321851 


1048 


10-678149, 


-009355 


43 9-990645 


1 9 • 


52 


9-313097 1003 


•686903 


' 9-322479 


1047 


10-677-521 


-009382 


45 ; 9 -9906 18 


! 8 


53 


9-313698 1001 


-686302 


9-323106 


1045 


10-676894 


-009409 


45 i9-990591 


1 7 


54!! 9-314297 1000 


•685703 


9-323733 


1044 


10-676267! 


-009435 


43 '9-990565 


6 


55 


1 9-314897 9y8 


•685103 


9-324358 


1043 


10-675642 


-009462 


45 9-990538 


! 5 


56 


i 9-315495 997 


•684505 


19-324983 


1041 


10-675017, 


-009489 


45 9-990511 


! 4 


57 


9-316092 99(3 


•683908 


9-325607 


1040 10-674393! 


-009515 


43 9-990485! 3 


58 


ll 9-316689 994 


•683311 


9-326231 


1039 10-673769 


•009542 


45 !9-990458;l 2 


59 |! 9-317284 \)'.ro 


•68271611 9-326853 


1037 10-673147! 


•009569 


45 j9-99043l 


1 


60;! 9-317879 ^)V)1 


-682121 19-327475 


lOoG 10-672525 ! 


-009596 


45 19-990404 


[l 


' 11 Cusiue. 


,>efuiit. Il C'>tangeu:. I'angent. Cosecant. 


j Sine. 



78 DEii. 



LOGARITHMIC SINES, ETC. 



53 



12 


DEG, 




















' 


Sine. 


Diff. 

100" 


Dosecant. 


Tangent. 


^^,J j Cotangent. 


Secant. 


Ditt. 

100" 


Cosine. 


' 





9-317879 




682121 


9-327474 


!lO-672526 


-009596 




9-990404 


60~ 


1 


9-318473 


990 


681527 


9-328095 


1036 10-671905 


-009622 


43 


9-990378 


159 


2 


9-319066 


988 


680934 


9-328715 


1038 10-671285 


•009649 


45 


9-990351 


%58 


3 


9-319658 


987 


680342 


9-329334 


1032 10-670666 


-009676 


46 


9-990324 


57 


4' 


9-320249 


986 


679751 


9-329953 


1030ilO-670047 


-009703 


46 


9-990297 


56 


5 


9-320840 


984 


679160 


9-330670 


1029110-669480 


•009730 


45 


9-990270 


66 


6 


9-321430 


983 


678570 


9-331187 


1028 


10-668813 


-009757 


45 


9-990243 


54 


7 


9-322019 


982 


677981 


9-331803 


1026 


10-668197 


-009785 


47 


9-990215 


53 


8 


9-322607 


980 


677393 


9-332418 


1025 


10-667682 


•009812 


45 


9-990188 


52 


9 


9-323194 


979 


676806 


9-333033 


1024 


10-666967 


-009839 


46 9-990161 


51 


10 


9-323780 


977 


676220 


9-333646 


1023 


10-666354 


•009866 


46 9-990134 


50 


11 


9-324366 


976 


675634 


9-334259 


1021 


10-665741 


•009893 


46 9-990107 


49 


12 


9-324950 


975 


675050 


9-334871 


1020 


10-665129 


•009921 


47 9-990079 


48 


13 


9-325534 


973 


674466 
673883 


9-335482 


1019 


10-664518 


•009948 


45 9-990052 


47 


14 


9-326117 


972 


9-336093 


1017 


10-663907 


•009976 


45 9-990026 


46 


15 


9-326700 


970 


673300 


9-336702 


1016 


10-663298 


•010003 


47 


9-989997 


45 


16 


9-327281 


969 


672719 


9-337311 


1016J10-662689I 


•010030 


45 


9^989970 


44 


17 


9-327862 


968 


672138 


9-337919 


1018 


10-662081 


•010068 


47 


9^989942 


43 


18 


9-328442 


966 


671558: 


9-338527 


1012 


10-661473 


•010086 


45 


9^989916 


42 


19 


9-329021 


965 


670979 


9-339133 


1011 


10-660867 


•010113 


47 


9-989887 


41 


20 


9-329599 


964 


670401 


9-339739 


1010 


10-660261 


•010140 


45 9-989860 


40 


21 


9-330176 


962 


669824 


9-340344 


1008 


10-659656 


•010168 


47 9-989832 


39 


22 


9-330753 


961 


669247 


9-340948 


1007 


10-659052 


•010196 


47 9-989804 


88 


23 


9-331329 


960 


668671 


9-341552 


1006 


10-658448 


•010228 


46 9-989777 


37 


24 


9-331903 


958 


668097 


9-342155 


1004 


10-067845 


•010251 


47 9-989749 


36 


25 


9-332478 


957 


667522 


9-342757 


1003 


10-657243 


•010279 


47 :9-989721 


86 


26 


9-333051 


956 


666949 


9-343358 


1002 


10-656642 


•010307 


47 9-989693 


34 


27 


9-333624 


954 


666376 


9-343958 


1001 


10-666042 


•010335 


47 9^989665 


33 


28 


9-334195 


953 


665805 


9-344558 


999 


10-655442 


•010363 


47 9^989637 


32 


29 


9-334767 


952 


665233 


9-345157 


998 


10-654843 


•010390 


45 9^989610 


31 


30 


9-335337 


950 


664663 ! 


9-345755 


997 


10-654245 


•010418 


47 9^989682 


30 


31 


9-335906 


949 


664094 1 


9-346358 


996 


10-653647 


•010447' 48 9^989553 


29 


32 


9-336475 


948 


663525 


9-346949 


994 


10-653051 


•0104751 47 !9^989525 


28 


33 


9-337043 


946 


662957 


9-347545 


993 


10-652465 


•010503 


47 !9-989497 


27 


34 


9-337610 


945 


662390 


9-348141 


992 


10-661869 


•010531 


47 


9-989469 


26 


35 


9-338176 


944 


661824 


9-348735 


991 


10-651266 


•010559 


47 


9-989441 


25 


36 


9-338742 


943 


661258 


9-349329 


990 


10-6.50671 


■010587 


47 


9-989413 


24 


37 


9-339307 


941 


660693 


9-349922 


988 


10-650078 


-010615 


47 


9-989386 


23 


38 


9-339871 


940 


660129 


9-350514 


987 


10-649486 


•010644 


48 


9-989356 


22 


39 


9-340434 


939 


659566 


9-351106 


986 


10-648894 


•010672 


47 !9^989828 


21 


40 


9-340996 


937 


659004 


9-361697 


986 


10-648303 


•010700 


47 |9^989800 


20 


41 


9-341558 


936 


658442 


9-352287 


983 


10-647718 


•010729 


48 19 -989271 


19 


42 


9-342119 


935 


657881 


9-352876 


982 


10-647124 


•010757 


'47 i 9 -989243 


!18 


43 


9-342679 


934 


667321 


1 9-353466 


981 


10-646685 


-010786 


48 i9 -989214!! 17 


44 


9-343239 


932 


656761 


i 9-354053 


980 


10-645947 


•010814 


47!9-989l86'!16 


45 


9-343797 


931 


656203 


9-354640 


979 


10-645360 


•0108431 48 


9^989157 


16 


46 


9-344355 


930 


655645 


9-36^227 


977 


10-644773 


•010872 48 


9^989128 


14 


47 


9-344912 


929 


655088 


9-355813 


976 


10-644187 


•OIO9O0I 47 


9^989100 


13 


48 


9-345469 


927 


654531 


9-356398 


976 


10-643602 


•010929 48 


9^989071 


12 


49 


9-346024 


926 


653976 


9-356982 


974 


10-643018 


•0109581 48 


9-989042 


11 


50 


9-346579 


925 


653421 


9-357566 


973 


10-642434 


•010986! 47 


9^989014 


10 


61 


9-347134 


924 


652866 


9-368149 


971 


10-641861 


•011016! 48 


9^988985 


9 


52 


9-347687 


922 


652313 


9-358731 


970 


10-641269 


•011044! 48 


9 •988956 


8 


63 


9-348240 


921 


651760 


9-359313 


969 


10-640687 


•011073 48 


9^988927 


7 


54 


9-348792 


920 


651208 


9-359893 


968 


10-640107 


•011102 48 


9^988898 


6 


55 


9-349343 


919 


650657 


9-360474 


967 


10-639526 


•011131148 


9^988869|i 5 


56 


9-349893 


917 


650107 


9-361053 


966 


10-638947 
10-638368 1 


•011160 48 


9^988840 i 4 


57 


9-350443 


916 


649557 


9-361632 


966 


•011189; 48 


9-988811:; 3 


58 


9-350992 


915 


-649008 


9-362210 


963 


10-637790 


•011218 48 


9-9887821 2 


59 


9-351540 


914 


•648460 


9-362787 


962 


10-637213 


•011247 48 


9-988753 1 


60 


9-352088 


913 


647912 


9-363364 

1 Cotangent. 


961 


10-636636 


•011276 48!9-988724' 


1 ' 


Cosiuo. 


1 Secant. 




Tangent. Cosecant. \ Sine. 1 ' , 



77 DKG 



54 



LOGARITHMIC SINES, ETC. 



13 


DEC. 




















' 


Sine. 


Diff. 

100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent, i Secant. 


Diff 

m' 


Cosine. 1 


■i 





9-352088 




•647912 


9^363364 




10-636636 


•011276 




9^988724| 


60 


1 


9-352635 


911 


•647365 


9-363940 


960 '10-636060 


•011305 


48 


9-988695 


59 


2 


9-353181 


910 


•646819 


9-364515 


959 10-63-5485 


•011334 


48 


9-988666|58 1 


3 


9-353726 


909 


•646274 


9 •36.5090 


958 ,10-634910 


•011364 


50 


9-988636i.57 


4 


9-354271 


908 


•645729 


9-365664 


957 10-634336 


•011393 


48 


9-988607;!-56 ! 


5 


9-354815 


907 


•645185 


9-366237 


955; 10-633763 


•011422 


48 


9-988578:155 


6 


9-355358 


905 


•644642 


9-366810 


954 '10-633190 


•0114.52 


50 


9 -988.5481154 j 


7 


9-355901 


904 


•644099 


9-367382 


953 110-632618 


-011481! 48 


9-988519.153 1 


8 


9 -3564431 903 


•643557 


9-367953 


9.52:10-632047 


•011511150 


9-988489:52 ! 


9 


9-356984 


902 


•643016 


9-368524 


951 10-631476 -011540148 


9-988460151 i 


10 


9 -307524 


901 


•642476 


9^369094 


950 10-630906 -011570, 50 


9-9884301.50 \ 


11 


9-358064 


899 


•641936 


9-369663 


949 10-630337 -011.599,48 


9-988401 149 ; 


12 


9-358603 


898 


•641397 


9-370232 


948 10-62&768 -011629150 


9-988371 48 ; 


13 


9-359141 i 897 


•640859 


9-370799 


946 10-629201 -0116.58148 


9-988342 ;47 


14 


9-359678 


896 


•640322 


9-371367 


945 ilO-628633, -0116881 50 


9-9883121,46 


15 


9-360215 


895 


•639785 


9-3719331 944 110-628067 


-011718; 50 


9-9882821145 


16 


9-360752 


893 


•639248 


9-372499 


943: 10-627.501 


•011748 50 


9-988252144 1 


17 


9-361287 


892 


•638713 


9-373064 


942! 10-626936 


•011777 48 


9^988223i43 1 


18 


9-361822 


891 


•638178 


9-373629 


941 i 10-626871 


•0118071 50 


9^988193:i42 


19 


9-362356 


890 


•637644 


9-374193 


940 i 10-625807 


-0118371 50 


9^9881G3l41 


20 


9-362889 


889 


•637111 


9-374756 


939 10-62.5244 


-0118671 .50 


9^988133l40 


21 


9-363422J 888 


•636578 


9^375319 


938 10-624681 


•011897 


50 


9-98810339 


22 


9-363954 


887 


•636046 


9-375881 


937 :10-624119 


-011927 


50 


9-988073 J38 


23 


9-364485 


885 


•635515 


9-376442 


935 10-623.558 •0119.57 


50 


9-988043:37 


24 


9-365016 


884 


•634984 


9-377003 


934110-622997 


•011987 


50 


9-988013i;.36 1 


25 


9-865546 


883 


•634454 


9^377563 


933 ! 10-622437 


-012017 


50 


9-987988135 


26 


9-366075 


882 


•633925 


9 •3781 22 


932il0-621878 


-0120471 50 


9-987953^134 


27 


9-366604 


881 


•633396 


9^378681 


931 110-621319 


1-012078 


52 


9-987922;!33 


28 


9-367131 


880 


•632869 


9-379239 


9.30jl0-620761 


-012108 


50 


9-987892;i32 


29 


9-367659 


879 


•632341 


9-379797 


929 ilO-620203 


1-012188 


50 


9^987862i!31 
9-9878.32 30 


30 


9-368185 


878 


•631815 


9-380354 


928|10-619646 


i-012168!50 


31 


9-368711 


876 


•631289 


9-380910 


927 110-619090 


•0121991 52 


9-987801i|29 


32 


9-369236 


875 


•630764 


9-381466 


926 


10-618.534 


1-012229: 50 


9-987771128 


' 33 


9369761 


874 


•630239 


9-382020 


925 


10-617980 


r012260| 52 


9-987740 


27 


34 


9-370285 


873 


•629715 


9^382575 


924 


10-617425 


{•012290; 50 


9-987710 


28 


35 


9-370808 


872 


•629192 


9-383129 


923 


10-616871 


1-012321 52 


9-987679 


25 


36 


9-371330 


871 


•628670 


9-383682 


922 


10-616318 


1-0123.51 50 


9-987649 


24 


37 


9-371852 


870 


•628148 


9^384234 


921 


10-61.5766 


1-0128821 .52 


9-987618 


28 


38 


9-372373 


869 


•627627 


9^384786 


920 


10-615214 


•012412:50 


9-987588 


22 


39 


9-372894 


867 


•627106 


9-385337 


919 


10^614663 


•01 2443 1 .52 


9-987.557 


21 


40 


9-373414 


866 


•626586 


9-385888 


918 


10-614112 


•0124741 52 


9-987526 


20 


41 


9-373933 


865 


•626067 


9-386438 


917 


10-613562 


•012.504 50 


9-987496 


19 


i 42 


19-374452 


864 


•625548 


9-386987 


916 


10-613013 


•012.535 52 


9-987465 


18 


i 43 


9-374970 


863 


•625030 


9-387536 


914 


10-612464 


•012-566 52 


9 -9874341 17 


44 


{9-375487 


862 


•624513 


9-388084 


913 


10-611916 


-0125971 52 


9-987403,16 


45 


19-376003 


861 


•623997 


9-388631 


912 


10-611369 


-012628j52 


9-987372i'l5 


46 


19-376519 


860 


•623481 


9-389178 


911 


iO-610822 


-0126-59 52 
-012690! 52 


9-987341j^l4 


I 47 


9-377035 


859 


•622965 


9-389724 


910|10-610276 


9-987310'13 


1 48 


9-377549 


858 


•622451 


9-390270 


909 110-609730 


•012721!. 52 


9-987279|!]2 


49 


19-378063 


857 


•621937 


9-390815 


908 1 10-609185 


•0127521 52 


9 -987248;! 11 


50 


9-378577 


856 


•621423 


9-391360 


907 110-608640 


•012788! -52 


9-987217!|10 


51 


9-379089 


854 


•620911 


9-391903 


906 


10-608097 


-0128141 52 


9-987186!! 9 


1 52 


9-379601 


853 


•620399 


9^392447 


905 


10-6075.53 


-012845 52 


9-987155;i 8 


53 


9-380113 


852 


•619887 


9^392989 


904 


10-607011 


■012876 


52 


9-987124i 7 


54 


9-380624 


851 


•619376 i 9-393631 


903 


10-606469 


•012908 


58 


9-9870921 


6 


55 


9-381134 


850 


-618866 i 9-394073 


902 


10-605927 


•012939 


52 


9-98706li 


5 


56 


9-381643 


849 


•618357 1 9-394614 


901 


10-605886 


•012970 52 |9^987030l 


4 


57 


9-382152 


848 


•617848!! 9-395154 


900 


10-604846 


•013002 .58 l9^986998'| 3 


58 


19-382661 


847 


•617339 !: 9-305694 


899il0-60480G! 


-013083; 52 !9-98G967;j 2 


1 S9 


19-383168 


846 


•616832 ! 9-396283 


898! 10-6087671 


•018064152 i9-98698()' 


1 


(50!' 9-383675 


845 


•616325 


9-8967711 897 ilO-608229 !|-()18i)96 


.•-8 |'.)-<»8»;^H)4' 





1 '. Cosine. i 


Se'jaur. Cotaii'ieac. ' 1'.i.,-.-ik C..s.= /.:> it 




.,!,... 


' 



76 UEG. 



LOGARITHMIC SINES, ETC. 



55 



14 


DE6. 




















i ' ^ 


Sine. ^; 


Cosecant. 


Tangent, i ^^^,; | Cotangent, j; Secant. ^^^^; 


Cosine, jl ' j 


0, 


9 -083670 1 


616325 


9-396771 1 10-603229 


-013096; 


9^986904; 60 | 


1; 


9-384182 


844' 


615818 


9-397309 896 10-6026911 


•013127 52 


9^986873 69 


^i 


9-384687 


843' 


615313 


9-3978461 896 10-602154 , 


-013159 53 19-986841 58 i 


s\ 


9-385192 


842; 


614808 


9-398383i 895 10-601617 ! 


•013191 53 


9-986809 57 \ 


4| 


9-385697 


841 1 


614303 


9-398919| 894 10-601081 i 


-013222 52 


9-986778 56 


5 


9-386201 


840 


613799 


9 •3994-55: 893 10-600545 ! 


-0132.54 53 


9-986746 55 


6 


9-386704 


839 


613296 


9-399990! 892 10-6000101 


-013286 53 19-986714 54 | 


7 


9-387207 


838 


612793 


9-400.524| 891 10-599476 i 


•013317 -52 


9-986683 53 ! 


8 


9-387709 


837 


612291 


9-4010-58 890 10-598942 


•013349 53 


9-986651 52 i 


9 


9-388210 


836 


611790 


9-401591' 889 10-598409; 


•013381 53 


9-986619 61 | 


10 


9-388711 


835 1 


611289 


9-402124! 888 10-597876! 


-013413; 53 


9-986-587 50 i 


11 


9-389211 


834 


610789 


9-402656! 887 10-597344! 


-013445 53 


9 •986.5.55 49 


12 


9-389711 


833 


610289 


9-403187; 886 10-596813 j 


•013477! 53 


9 •986-523 ;48 


13 


9-390210 


832 


609790 


9-403718^ 885 '10-5962821 


•013-5091 -53 


9^986491 147 


14 


9-390708 


831 


609292 


9-40424yi 884 10-595751 i 


-0135411.53 


9^986459 |46 


151 9-391206 


830 


608794 


9-404778' 883 10-5952221 


-013573J 63 


9-986427:45 


16! 9-391703 


828 


608297 


9-405-30b: 882, 10-594692 


-0136051 53 


9^986395 44 


17^ 9-392199 


827 


607801 


9-405836; 881 :i0-594164 


•0136371 53 


9-986363 :43 


18 1 9-392695 


826 


607305 


9-406364! 880 I10-593636 


•0136691 53 


9-98633i;i42 


19! 9-393191J 825 


606810 


9-406892^ 879 10-593108! 


•013701! 63 


9 -986299! '41 


20 9-393685 824 


606315 


9-407419 878 10-592.581 


-013734 55 


9-986266 40 


21 9-394179! 823 


605821 


9-407945 877 ilO-592055 


-0137661 53 


9-986234! 39 


22 ■19-394673! 822 


605327 


9-408471 876 10-591529! 


•013798! 53 


9-986202! 38 


23 19-395166 821 


604834 


9-408997! 875 10-.591003 1 


-013831 55 


9-986169 37 


24 !i 9-395658! 820 


604342 


9-409-5211 874 ■10-590479! 


-013863153 9-986137,36 | 


25:19-3961501 819 


603850 


9-410045 874!10-5899-55! 


-013896! .55 9-986104 35 i 


26 ,[9-396641 818 


603359 


9-410569| 873!10-589431 


•013928| 53 


9-986072 .34 


27;; 9-397132! 817 


602868 


9-411092! 872 110-588908 


•013961 .55 


9-^86039 33 1 


28: 9-397621! 817 


602379 


9-411615; 871 !l0-588385 


-013993 


53 


9-986007 32 j 


29 '9-398111! 816 


601889 


9-412137, 870 10-587863 


-014026 


55 


9-98-5974 31 | 


30 9-398600 815 


601400 


9-412658! 869 ilO-587342 


•014058 


53 


9-985942 :30 1 


81;! 9-399088! 814 


600912 


9-413179 868! 10 -.586821 


•014091 


55 


9-98-5909J29 | 


82 19 -399575 i 813 


600425 


9-413699 867 1 10-586301 


•014124 


55 


9-985876'i28 | 


88!|9-400062i 812 


599'.:)38 


9-414219 866! 10-58.5781 


•014157 55 


9-985843-;,27 i 


34 ! 9-400-5491 811 


599451 


9-414738! 866 !l0-68.5262 


•014189 


53 


9-98581 ll!26 


85 ,19-401035; 810 


598965 


9-415257: 864 ilO-584743 


•014222 


66 


9-985778!!26 


86: 9-401520! 809 


598480 


9-415775 864il0-584225 


-014255 


65 


9-9857461124 


37' 9-402005! 808 i 


597995 


9-416293 863 '10-583707 


-014288 


56 


9-985712il23 


88;! 9-402489! 807 i 


597511 


9-416810 862; 10 -.583 190 


-014321 


55 


9-985679122 


39 


9-402972 806! 


597028 


9-417326! 861 ;10-582674 


•0143-54 


65 


9-98564621 


40 


9-403455! 8U5 I 


596545 


9-417842! 860 


10-582158 


•014387 


55 


9-986613il20 I 


41 


9-403938 


804 


596062 


9-418-3-58, 859 


10-581642 


■014420 


5519-986580 


19 


42 


9-404420 


803 


595580 


9-418873! 858 


10-581127 


-014453 


56 


9-985547 


18 


43 


9-404901 


802 


595099 1:9-41 9387; 857 


10-580613 


•014486 


65 


9-985514 


17 


44 


9-405382 


801 


594618 ■! 9-410901! 856 110-580099 


•014.520 


57 


9-985480 


16 


45 


9-405862 


800 


594138 i 9-420415 855 ! 10-579585 


•014553 


55 


9^985447 


15 


46 


9-406341 


799 


593659 '! 9-420927, 855 |l0-579073 


•014586 


55 


9-985414 


14 


47 


9-406820 


798 


593180.;! 9-421440 854 


10-678560 


-014619 


65 


9-985381 


13 


48 


9-407299 


797 


592701 i 9-421952 853 


10-578048 


-014653 


57 1 9 •985347 


12 


49 


9-407777 


796 


592223 :; 9-422463 852 


10-577537 


•014686 


55 19.985314 


11 


50 


9-408254 


796 


591746 119-422974 851 


10-577026 


•0147201 57 ;9-985280 


10 


51 


9-408731 


794 


591269!! 9-423484 850 


10-576516 


•014753; 65 !9^985247 


9 


52 


9-409207 


794 


590793 1 9-423993 849 10-576007 


•0147871 57 19-985213 


8 


63 


9-409682 


793 


590318 ! 9-424-503; 848 ! 10-575497 


•014820! 55 19-985180 


7 


54 


9-410157 


792 


589843 1! 9-42-5011 


848 10-574989 


-014854! 57!9-985146 


6 


55 


9-410632 


791 


5893681! 9-42-5519 


847 ilO-674481 


•014887! 55 ;9-985113 


6 


56 


9-4111061 790 


588894 !' 9-426027 


846 


10-573973 1 


•014921' 57 9-986079 


4 . 


57 1 


9-411579' 789 


588421 ,! 9-426634 846 


10-573466 ' 


•014965' 57 9 •985045 


3 ! 


68 


9-412052! 788 


587948!! 9-4270411 844 


10-572959 


•014989 57 9^985011 


2 


59 1 9-412524 


787 


587476 9-427547! 843 


10-572453 


•016022! 55 l9-<»84978'! 1 i 


60: 


9-412996 


786 


587004 ! 9-428052 843 


10-571948 -0150561 57 9-984944 


i 

-1 


'■ 


Cosine. 




Secant. 


Cotangent. 




Tangent. 


Cosecant. 




Sine. 



75 DKG. 



56 



LOGARITHMIC SINES, ETC. 



15 DEG. 



' 


j Sine. 


Diff. 
100" 


Cosecant. Tangent. 


Diff. 
100" 


Cotangent. 


Secant. ^^^^^,^\ Cosine. 


1' 





9-412996 




-587004 9-428052 




10-571948 


-0150561 !9-984944|i60 


1 


9-413467 


785 


-586533 9-428557 


842 


10-571443 


-015090, 57j9-984910!l59 


2 


9-413938 


784 


•586062 


9-429062 


841 


10-570938 


•015124: 57 ,9 -984876: :58 


3 


1 9-414408 


783 


•585592 ; 


9-429566 


840 


10-570434 


•0151581 57 i9-984842!|57 


4 


'9-414878 


783 


•585122 


9-430070 


839 


10-569930 


•015192 57 19-984808 156 


5 


j 9-415347 


782 


•584653 • 


9-430573 


838 


10-569427 


•015226 57 19-984774 ,55 


6 


19-415815 


781 


•584185 ' 


9-431075 


838 


10-568925 


•0152601 57 9-984740 154 


7 


1 9-416283 


780 


•583717 


9^431577 


837 


10-568423 


-015294; 57 '9-984706 [53 


8 


19-416751 


779 


•583249 


9^432079 


836 


10-567921 


-015328; 57 9-984672 |52 


9 


19-417217 


778 


•582783 


9-432580 


865 


10-567420 


•015362] 57 9-984638 51 


10 


9-417684 


777 


•582316 


9-433080 


834 


10-566920; 


-0153971 58 9-984603:50 


nil 9-418150 


776 


•581850 


9-433580 


833 


10-566420 i 


•015431! 57 9-984569 49 


12 ; 9-418615 


775 


-581385 


9-434080 


832 


10-565920 ; 


•015465; 57 9-984535, [48 


13 I 9-419079 


774 


•580921 


9-434579 


832 


10-565421 


•015500 58 9-984500; 47 


14 9-419544 


773 


-580456 , 


9-435078 


831 


10-564922 1 


•015534 57 9-984466 46 


15N-420007 


773 


•579993 


9-435576 


830 


10-564424 


-015568 57 9-984432 |45 


16 J! 9-420470 


772 


•579530 


9-436073 


829 


10-563927 


-015603 


58|9-984397i(44 


17 !i 9-420938 


771 


•579067 


9-436570 


828 


10-563430 


-015637 


57 19-984368:143 


18 i 9-421395 


770 


•578605 


9-437067 


828 


10-562933 


•015672 


58 ;9-984328'!42 


19 i 9-421857 


769 


•578143 


9-437563 


827 


10-562437 ! 


•015706 


58 ;9 -984294 ;41 


201:9-422318 


768 


•577682 


9-438059 


826 


10-561941 


•015741 


57 |9-984259,[40 


21: 9-4227781767 


•577222 


9-438554 


825 


10-561446 


•015776 


58 [9-9842241:39 


22: :9-423238i 767 


•576762 


9-439048 


824 


10-560952 


•015810 


57 |9-984190 38 


23 il 9-423697 


766 


•576303 


9-439543 


823 


10-560457 


•015845 


58 i9^984155:;37 


24! 9-424156 


765 


•575844 


9-440036 


823 


10-559964 1 


-015880 


58 19-984120 136 


251:9-424615 


764 


•575385 


9-4405291 822 


10-559471 ! 


•015915 


58 19-984085' 35 


26 9-425073 


763 


•574927 


9-441022 821 


10-558978 


•015950 


58 9-984050; 34 


27 ''9-4255301 762 


•574470 


9-44] 51 4 


820 


10-558486 


-015985 


.58 9-984015 133 i 


28:19-425987! 761 


•574013 


9-442006 


819 


10-557994 


-016019 


57 19-983981 


32 


29;: 9-426443, 760 


•573557 


9-442497i 819 


10-557503 


-016054 


58 j9-983946 


31 


30;! 9-426899! 760 


•573101 


9-442988 818 


10-557012 


-016089 


58 9-983911 


30 


31: 9-427354 759 


-572646 


9-443479) 817 


10-556521 


-016125' 60 '9-983875' 29 


32 ,9-427809; 758 


-572191 


9-443968; 816 10-556032 


•OI6I6O; 58 i9-983840,[28 


33 9-428263 7.37 


•571737 


9-444458 816 10-555542 


-OI6I95I 58 9-983805['27 


34 9-428717 756 


-571283 


9-4449471 815 10-555053 


-0162301 58 i9-983770:;26 


351 9-429170 755 


•570830 


9-445435t 814 10-554565 


,-0162651 58 19-983735:25 


36119-429623 754 


•570377 


9-445923 813 10-554077 


-016300: 58 |9-983700ii24 


37119-430075 753 


•569925 


9-446411 812 10-553589 


•016336: 60 i 9-983664^23 


38 9-430527 752 


-569473 


9-446898 812 10-553102 


-016371; 58:9-983629' 22 


39 9-430978 752 


•569022 


9-447384i 811 10-552616 


-016406 5819-983594 21 


40 i 9-431429 751 


•568571 


9-447870 


810 ilO-552130 


-016442' 60 i 9 -983558 20 


41 ; 9-431879 750 


•568121 


9-448356 


809 ilO-551644 


•01 6477 i 58 !9-983523i 19 


42 9-432329 749 


•567671 


9-4488411 809 10-551159 


-016513; 60 9-983487 !l8 


43 9-432778 749 


•567222 


9-449326 808 10-550674 


'-016548: 58{9-983452;jl7 
•016584 60 9-983416 16 


44 ,9-433226 748 


•566774 


9-449810 807 ,10-.550l90 


45 9-433675 747 


•566325 


9-450294 806 10-549706 


-016619 58 9-983381, Il5 


46 9-434122 746 


•565878 


9-4507771 806 10-549223 


-016655 60;9-983345 


14 


47 9-434569 745 


•565431 


9-451260! 805 10-548740 


-016691 60 '9-983309 


13 


48 9-435016 744 


■564984 


9-451743 


804 110-548257 


•016727 60 9^983273 


12 


49 9 -435462 744 


1 ^564538 


9-452225 


803 i 10 -547775 


•016762 58,9-983238 


11 


50 19-435908 743 


•564092 


9-452706 


802 10-547294 


,-016798 60 .9-983202 


10 


51 9-436353 742 


-563647 


9-453187 


802 [10-546813 


,•016834 60J9-983166 


9 


52:19-436798 741 


•563202 


9-453668 


801 ,10-546332 


;-0l6870 60 [9-983130 


8 


53' 9-437242 740 


•562758 


9-454148 


800 jlO-545852 


-016906 60|9-983094 


7 


54 1! 9-437686 740 


•562314 


9-454628 


799 110-545372 


-016942 60,9-983058 


6 


55'! 9-438129 "^^9 


j ^561871 


i 9-455107 


799 10-544893 


•016978 6019-983022 


5 


56:19-438572 738 


j ^561428 


9-455586 


798 110-544414 


•017014; 60 


9-982986 


4 


57 i 9-439014 737 


1 ^560986 


, 9-456064 


797 


10-543936 


•017050 


60 


9-982950 


3 


58 1 9-439456 736 


•560544 


9-456542 


796 


10-543458 


•017086 


60 


9^982914 


2 


59:19-439897 736 


1 ^560103 


: 9-457019 


796 


10-542981 


•017122 


60 


9-982878 


1 


60119-440338 735 


i -559662 


9-457496 


795 


10-542504 


-017158 


60 J9-982842 





' |l Cosine. 


1 Secant. 


Cotangent. 




Tangent. 


Cosecant. | 1 Sine. 



74 D£6. 



16 DEG. 



LOGARITHMIC SINES, ETC. 



' 


Sine. 


Diff. 
100" 


Cosecant. | 


Tangent. 


I^ff- ! Cotangent. 


Secant. 


^; Cine. ^ 


' 





9-440338 




-559662 1 


9-457496 


, I10-542504 


-017158 


9-982842|:60 


1 


9-440778 


734 


•559222 i 


9-457973 


794:10-542027 


-017195 


62 9-9828051:59 


2 


9-441218 


733 


-558782 1 


9-458449 


793 10-541551 


-017231 


60 9-9827691 58 


3 


9-441658 


732 


•558342 


9-458925 


793 10-541075 


-017267 


60 9-9827o3::57 


4 


9-442096 


731 


-557904 


9-459400 


792 ilO-540600 


-017304 


62 9-982096'5() 


si 


9-442635 


731 


-5574651 


9-459875 


791 


10-540125 


-017340 


60 9-982660:55 


6 


9-442973 


730 


-557027] 


9-460349 


790 


10-539651 


-017376 


60 9-982624 '54 


7 


9-443410 


729 


-556590 


9-460823 


790 


10-539177 


-017413 


62 9-982587:'53 


8 


9-443847 


728 


•556153; 


9-461297 


789 


10-538703 


-017449 


60 9 -98255 Ij 52 


9 


9-444284 


727 


•555716 i 


9-461770 


788 


10-538230 


-017486 


62 9-982514; 51 


10 


9-444720 


727 


•555280 \ 


9-462242 


788 


10-537758 


-017523 


62 :9-982477 50 


1^ 


9-445155 


726 


•554845 


9-462714 


787 


10-537286 


-017559 


60 9-982441 49 
62 9-982404148 


12 


9-445590 


725 


•554410 


9-463186 


786 


10-536814 


^17596 


131 


9-446025 


724 


•553975 j 


9-463658 


785 


10-536342 


•017633 


62 9-982367 47 


14 


9-446459 


723 


•553541 


9-464128 


785 


10-535872 


•017669 


60 9-982331 |46 


15 1 


9-446893 


723 


•553107 


9-464599 


784 


10-535401 


-017706 


62 9-982294: 45 


161 


9-447326 


722 


•552674 


9-465069 


783 


10-534931 


-017743 


62 9-982257|'44 


17 


9-447759 


721 


•552241 


9-465539 


783 


10-534461 


-017780 


62 9-982220 43 


18 


9-448191 


720 


-551809! 


9-466008 


782 


10-533992 


-017817 


62 9-982183'42 


19 


9-448623 


720 


-551377 


9-466476 


781 


10-533524 


-017854 


62 9-982146|:41 


20 


9-449054 


719 


•550946 1 


9-466945 


780 


10-533055 


-017891 


62 9-982109 


40 


21 


9-449485 


718 


•550515 


9-467413 


780 


10-532587 


-017928 


62 9-982072! 


39 


22 


9-449915 


717 


-550085 1 


9-467880 


779 


10-532120 


-017965 


62 9-982035 


38 


23 


9-450345 


716 


-549655 i 


9-468347. 


778 


10-531653 


-018002 


62 9-981998 


37 


24 


9-450775 


710 


-549225 i 


9-468814 


778 


10-531186 


-018039 


62 9-981961| 


36 


25 


9-451204 


715 


•548796 


9-469280 


777 


10-530720 


-018076 


62 9-981924 


35 


26 


9-451632 


714 


-548368 1 


9-469746 


776 


10-530254 


•018114 


63 9-9818861 34 


27 


9-452060 


713 


•5479401 


9-470211 


775 


10-529789 


-018151 


62 9-981849: 33 


28 


9-452488 


713 


-54751211 9-470676 


775 


10-529324 


-018188 


62 9-981812; 


32 


29 


9-452915 


712 


-5470851 9-471141 


774 


10-528859 


-018226 


63 9-981774 


31 


30 


9-453342 


711 


-546658 ji 9-471605 


773 


10-528395 


-018263 


62 9-981737! 


30 


31 


9-453768 


710 


•54G232ll 9-472068 


773 110-527932 


•018300 


62 9-981700! 


29 


32 


9-454194 


710 


-545806 II 9-472532 


772 ilO-527468 


-018338 


63 


9-981662 


28 


33 


9-454619 


709 


•545381 1 9-472995 


771 110-527005 


-018375 


62 


9-981625 


27 


34 


9-455044 


708 


-544956 'i 9-473457! 771 ilO-526543 


-018413 


63 


9-981587 


26 


35 


9-455469 


707 


•544531 jl9^473919 


770 ilO-526081 


-018451 


63 


9-981549 


25 


36 


9-455893 


707 


•544107 i 9^474381 


769 110-525619 


•-01 8488 


62 


9-981512 


24 


37 


9-456316 


706 


•543684 i! 9-474842 


769 


10-525158 


-018520 


63 


9-981474' 


23 


38 


9-456739 


705 


•543261 1| 9-475303 


768 


10-524697 


-018564 


63 


9-981436 


22 


39 


9-457162 


704 


-542838' 9-475763 


767 


10-524237 


-018601 


62 


9-981399 


21 


40 


9-457584 


704 


'542416 ) 9-476223 


767 


10-523777 


-018639 


63 


9-981361 


20 


41 


9-458006 


703 


•541994!! 9-476683 


766 


10-523317 


-018677 


63 


9-981323 


19 


42 


9-458427 


702 


•541573 II 9-477142 


765 


10-522858 


-018715 


63 


9-981285 18 


43 


9-458848 


701 


•541152 9-477601 


765 


10^522399 


-018753 


63 


9-981247 17 


44 


9-459268 


701 


•540732 li 9^478059 


764 


10^521941 


-018791 


63 


9-981209 16 


45 


9-459688 


700 


•540312 Ii9^478517 


763 


10^521483 


-018829 


63 


9-981171 


15 


46 


9-460108 


099 


•539892 !| 9^478975 


763 


10^521025 


-018867 


63 


9-981133 


14 


47 


9-460527 


698 


•539473 il 9^4794321 762 1 10^520568 


1-018905 


63 


9-981095 


13 


48 


9-460946 


698 


•539054 


9 -479889 j 761 


10-520111 


•018943 


63 


9-981057 


12 


49 


9-461364 


697 


•538636 


' 9-480345! 761 


10-519655 


-018981 


63 


9-981019 


11 


50 


9-461782 


696 


-538218 


19-480801 


760 


10-519199 


-0190191 63 


9-980981 


10 


51 


9-462199 


695 


-537801 


19-481257 


759 


10-518743 


; -019058; 65 19-980942 


9 


52 


9-462616 


695 


•537384 


9-481712|759 


10-518288 


! -019096 63 9-980904 


8 


53 


9-463032 


694 


•536968 


9-482167 758 


10-517833 


-019134 63 9-980866 


7 


54 


9-463448 


693 


•536552 


9 482621 757 


10-517379 


:-019173 65 9-980827 


() 


55 


9-463864 


693 


•536136 


9-483075 757 


10-516925 


-019211 63 '9-980789 


5 


56 


9-464279 


692 


•535721 


9-483529 756 


10-516471 


1-019250 65 9-980750 


4 


57 


9-464694 


691 


-535306 


19-4839821755 


10-516018 


-019288 63 9-980712 


3 


58 


9-465108 


690 


-534892 


1 9-4844351 755 


10-515565 


i-019327l 65 9-980673 


2 


59 


9-465522 


690 


•534478 


i 9 -4848871 754 


10-515113 


-019365 03 9-980635 


1 


60 


9-465935 


689 


•534065 


1 9-485339; 753 


10-514661 

1 


-019404' 65 .9-980596 

1 





Cosiae. 




Seoaut. 


! C^jtiiugeut. Tanjjeiit. Cosecant. Sine. 


i ' 



73 DE6. 



58 



LOGARITHMIC SINES, ETC. 



ir 


r)EG. 




















, 


Sine. 


Ditt-. 1 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


r-! cosine. 







9-465935 




•534065 


9-485339 




10-514661 


-019404 


!9^980596''60 


1 


9-466348 


688 


•533652 


9-485791 


753 


10-514209 


-019442 


63 '9-980558!.o9 


2 


9-466761 


688 


•533239 


9-486242 


752 


10-513758 


-019481 


65'9-9805l9i58 


3 


9-467173 


687 


•532827 


9-486693 


751 


10-513307 


■019520 


65 9 -980480 57 


4 


9-467585 


686 


-532415 


9-487143 


751 


10-512857 


-019558 


63 ,9-980442 56 


6 


9-467996 685 


-532004 


9-487593 


750 


10-512407 


•019597 


65 9-980403 55 


6 


9-468407i 685 


-531593 


, 9-488043 


749 


10-511957 


-019G36 


65 9-980364 54 


7 


9-468817| 684 


-531183 


9-488492J 749 


10-511508 


-019675 


65 9-980325 53 


8 


9-469227 


683 


-530773 


9-488941! 748 


10-511059 


•019714 


65 9-980286 52 


9 


9-469637 


683 


•530363 


' 9-489390' 747 


10-510610 


•019753 


65 9-980247 51 


10 


9-470046 


682 


•529954 


9^489838! 747 


10-510162 


-019792 


65 9-980208 50 ! 


11 


9-470455 


681 


•529545 


,9-490286: 746 


10-509714 


-019831165 9-980169 49 i 


12 


9-470863! 680 


•529137 


,9-490733: 746 


10-509267 


-019870 65 9-980130 48 1 


13 


9-471271 


680 


•528729 


9-491180 745 


10-508820 


•019909 65 


9-980091 47 


14 


9-471679 


679 


•528321 


'9-491627! 744 


10^508373 


-019948165 9-980052 46 i 


15 


9-472086! 678 


•527914 


9-492073' 744I1O-5O7927 


-019988 67 9-980012 '45 


16 


9-4724921 678 


•527508 


19-492519 743 10-507481 


-020027 65 9-979973 44 


17 


9-4728981 677 


•527102 


9-492965 743 |l(»-507035 


•020066 65 9-979934 43 


i 18 


9-473304 676 


•526696 


1 9^493410 742 


10-506590 


-020105; 65 9-979895 42 ! 


19 


9-473710l 676 


•526290 


: 9-493854; 741 


10-506146 


-020145 67 9-979855 41 1 


20 


9-474115 675 


•525885 


9-494299 740 


10-505701 


-020184 


65 


9-979816^40 


21 


9-4745191 674 


•525481 


9-494743 740 


10-505257 


•020224 


67 


9-979776 39 


i 22 


9-474923! 674 


•525077 


9-495186 739 


10-504814 


-020263 


65 


9-979737 38 


1 23 


9-475327 673 


■524673 


19-495630 739 


10-504370 


-020303 


67 


9-9796971 37 


1 24 


9-475730 672 


•524270 


9-496073^ 738 


10-503927 


-020342 


65 


9-979658:36 


25 


9-476133 672 


•523867 


9-496515^ 737 


10-503485 


-020382 


67 


9-979618 35 


26 


9-476536 671 


•523464 


9-496957 737 


10-503043 


-020421 


65 


9-979579 34 


27 


9-476938 670 


•523062 


9-497399 736 


10-502601 


•020461 


67 


9-979539;'33 


28 


9-477340 


669 


•522660 


9-497841 736 


10-502159 


-020501 


67 


9-979499:32 


29 


9-477741 


669 


•522259 


9-498282 735 


10-501718 


-020541 


67 


9-979459;31 


30 


9-478142 


668 


•521858 


9-498722: 734 


10-501278 


•020580 


65 


9-979420=30 


31 


9-478542! 667 


•521458 9-499163 734 


10-500837 


•020620 


67 


9-979380;i29 


32 


9-4789421 667 


•5210.58 9-499603 733 


10-500397 


•020660 


67 


9-979340 128 


33 


9-479342 666 


•520658 9-500042: 733 


10-499958 


•020700 


67 


9-979300 27 


34 


9-479741 


665 


-520259 9-500481 732 


10-499519 


1 -0207 40 


67 


9-979260 26 ! 


35 


9-480140 


665 


•519860 9-500920 731 


10-499080 


•020780 


67 


9^9792201 (25 


36 


9-480539 


664 


•519461 9-501359 731 


10-498641 


•020820 


67 


9^979180 124 


37 


9-480937 


663 


•519063 9-501797 730 


10-498203 


•020860 


67 


9^9791 40' !23 


38 


9-481334 


663 


•518666 9^502235' 730 


10-497765 


•020900 


67 


9^979100i22 


39 


9-481731 


662 


■ •518269 9-502672: 729 |l0^497328 


•020941 


68 


9-979059l'21 


40 


9-482128 


661 


•517872 9-503109; 728 


10-496891 


■020981 


67 


9^979019 20 ! 


41 


9-482525 


661 


•517475 9^503546; 728 


10-496454 


•021021 


67 


9^978979 


19 


42 


9-482921 


660 


•517079 9-503982; 727 


10-496018 


•021061 


67 


9-978939 


18 


43 


9-483316 


659 


•516684 9-504418! 727 


10-495582 


•021102 


68 


9 •978898 


17 


44 


9-483712 


659 


•516288 9-5048-54: 726 


10-495146 


•021142 


67 


9-978858 


16 


45 


9-484107 


658 


•515893 9-5052891 725 


10-494711 


•021183 


68 


9-978817 


15 


46 


9-484501 


657 


•51-5499 9-5057241 725 


10-494276 


•021223 


67 


9-978777 


14 


47 


9-484895 


657 


•515105 9-506159! 724 


10-493841 


•021263 


67 


9-978737 


Il3 


48 


9-485289 


656 


•514711 9^506593!724 


10^493407 


•021304 


68 


9-978696 


12 


49 


9-485682 


655 


•514318 9-507027! 723 


10^492973 


•021345 


68 


9-978655 


11 


60 


9-486075 


655 


-513925 9-507460; 722 |l0-492.540 


•021385 


67 


9-978615 


10 


51 


9-486467 


654 


•513533 9^507893 


722 110-492107 


•021426 


68 


9-978-574 


9 


52 


9-486860 


653 


•513140 9-508326 


721 


10-491674 


•021467 


68 


9-978533 


8 


53 


9-487251 


653 


•512749 9-508759 


721 


10-491241 


•0215071 67 


9^978493 


7 


54 


9-487643 


652 


•512357 9^509191 1720 


10-490809 


•021548 


68 


9-978452 


6 


55 


9-488034 


651 


•511966 9-509622 


719 


10-490378 


•021589 


68 


9^978411 


5 


56 


9-488424 


651 


•511576 9-510054 


719 


10-489946 


•021630 


68 


9-978370 


4 


57 


9-488814 


650 


•511186 9-510485 


718 


10-489515 


1-021671 


68 


9-978329| 


3 


58 


9-489204 


650 


•510796 9-51U916 718110-489084 


•021712 


68 


9-978288 


2 


1 59 


9 489593 


649 


•510407 9-511346! 717 10-488654 


N021753 


68 


9-978247 


1 


60 


9-489982 


648 


•510018 9-511776J 717 10-488224 


•0217941 68 


9-978206 





~^ 


Cosine. 


Secant. Cotangent. , Tangent. ' Cosecant.; 


Sine. ! 



72 DEG. 



LOGARITHMIC SINES, ETC. 



59 



IS 


DEG. 























Sine. 


Diff. 

100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


' 


9-489982 




-510018 


9-511776 




10-488224 


-021794 




9^978206 


6O' 


1 


9-490371 


648 


•509629 


9-512206 


716 


10-487794 


-021835 


68 


9^978165 


59 


2 


9-490759 


648 


•509241 


9-512635 


716 


10-487365 


-021876 


68 


9-978124-58 


3 


9-491147 


647 


-508853 


9-513064 


715 


10-486936 


-021917 


68 


9-978083!l57 


4 


9-491535 


646 


•508465 


9-513493 


714 


10-486507 


•021958 


69 


9-978042|56 


5 


9-491922 


646 


•508078 


9-513921 


714 


10-486079 


-021999 


69 


9-978001 55 


6 


9-492308 


645 


•507692 


9-514349 


713 


10-485651 


-022041 


69 


9-977959L54 


7 


9-492695 


644 


•507305 


9-514777 


713 


10-485223 


-022082 


69 


9-97791853 


8 


9-493081 


644 


•506919 


9-51.5204 


712 


10-484796 


-022123 


69 


9-977877 52 


9 


9-493466 


643 


•506534 


9-515631 


712 


10-484369 


-022165 


69 


9-977835I51 


10 


9-493851 


642 


•506149 


9-516057 


711 


10-483943 


-022206 


69 


9-977794j50 


11 


9-494236 


642 


•505764 


9-516484 


710 


10-483516 


-022248 


69 


9^977752|149 


12 


9-494621 


641 


•505379 


9-516910 


710 


10-483090 


-022289 


69 


9-97771l!|48 


13 


9-495005 


641 


•504995 


9-517335 


709 


10-482665 


-022331 


69 


9-977669il47 


14 


9-495388 


640 


•504612 


9-517761 


709 


10-482239 


-022372 


69 


9-977628i;46 


15 


9-495772 


639 


•504228 


9-518185 


708 


10-481815 


•022414 


69 


9-9775861I45 


16 


9-496154 639 


•503846 


9-518610 


708 


10-481390 


•02245G 


69 


9-9775441:44 


17 


9-496537 


638 


•503463 


9-519034 


707 


10-480966 


-022497 


70 


9 -9775031143 


18 


9-496919 


637 


•503081 


9-519458 


706 


10-480542 


-022539 


70 


9-97746li42 


19 


9-497301 


637 


•502099 


9-519882 


706 


10-480118 


-022581 


70 


9-977419ii41 


20 


9-497682 


636 


•502318 


9-520305 


705 


10-479695 


-022623 


70 


9-977377il40 


21 


9-498064 


636 


•501936 


9-520728 


705 


10-479272 


-022665 


70 


9-977335^39 


22 


9-4984441 635 


•501556 


9-521151 


704 


10-478849 


-022707 


70 


9-977293^38 


23 


9-498825 


634 


•501175 


9-521573 


704 


10-478427 


-022749 


70 


9-977251 37 


24 


9-499204 


634 


•500796 


9-521995 


703 


10-478005 


-022791 


70 


9-977209:'36 


25 


9-499584 


633 


•500416 


9 -.52241 7 


703 110-477583 


-022833 


70 


9-977167i35 


26 


9-499963 632 


•500037 


9-522838 


702 


10-477162 


-022875 


70 


9-977125 34 


27 


9-500342! 632 


.499658 


9-523259 


702 


10-476741 


-022917 


70 


9-977083 33 


28 


9-500721 


631 


•499279 


9-523680 


701 


10-476320 


-022959 


70 


9-977041i'32 


29 


9-501099 


631 


•498901 


9-524100 


701 


10-475900 


-02300] 


70 


9-976999 


31 


30 


9-501476 


630 


•498524 


9-524520 


700 


10-475480 


-023043 


70 


9-976957 


30 


31 


9-501854 


629 


•498146 


9-524939 


699 


10-475061 


-023086 


70 


9-976914 


29 


32 


9-502231 


629 


•497769 


9-5253.59 


699 


10-474641 


-023128 


70 


9-976872 


28 


33 


9-502607 


628 


•497393 


9-525778 


698 


10-474222 


-023170 


70 


9-976830 


27 


34 


9-502984! 628 


.497016 


9-.526197 


698 


10-473803 


-023213 


71 


9-976787 


26 


35 


9-503360 627 


•496640 


9-526615 


697 


10-473385 


•023255 


71 


9-976745 


25 


35 


9-503735! 626 


.496265 


9-527033 


697 


10-472967 


•023298 


71 


9-976702 
9-976660 


24 


! 37 


9-504110 


626 


.495890 


9-527451 


696 


10-472549 


•023340 


71 


23 

22 


38 


I 9-504485 


625 


.495515 


9-527868 


696 


10-472132 


•023383 


71 


9-976617 


39 


! 9-504860 


625 


.495140 


9-528285 695 


10-471715 


-023426 


71 


9-976574 


21 


1 40 


9-505234 


624 


.494766 


9-528702 


695 


10-471298 


-023468 


71 


9-976532 


20 


1 41 


9-505608 


623 


.494392 


9-529119 


6S4 


10-470881 


-023511 


71 


9-976489 


19 


42 


9-505981 


623 


•494019 


9^529535 


694 


10-470465 


•023554 


71 


9-976446 


18 


43 


9-506354 


622 


.493646 


9-529950 


693 


10-470050 


•023596 


71 


9-976404 


17 


44 


9-506727 


622 


.493273 


9-530366 


693 


10-469634 


•023639 


71 


9-976361 


16 


45 


19-507099 


621 


•492901 


9-530781 


692 


10-469219 


•023682 


71 


9-976318 


Il5 


46 


9-507471 


620 


.492529 


9-531196 


691 


10-468804 


•023725 


71 


9-976275'll4 1 


47 


19-507843 


620 


.492157 


9-531611 


691 


10-468389 


•023768 


71 


9-976232:13 | 


48 


j 9-5082141 619 


•491786 


'9-532025 


690 


10-467975 


•023811 


72 


9-976189 


12 1 


49 


' 9-508585 


619 


.491415 


1 9-532439 


690 


10-467561 


•023854 


72 


9-976146 


11 


50 


1 9-508956 


618 


.491044 


9-532853 


689 


10-467147 


•023897 


72 


9-976103 


10 


i SI 


i 9-509326 


618 


.490674 |! 9^533266 


689 


10 466734 


•023940 


72 


9-976060 


9 


1 52 


19-509696 


617 


•490304 19-533079 


688 


10-466321 


•023983 


72 


9-976017 


8 


53 


9-510065 


616 


•489935 ,9-534092 


688 


10^465908 


•024026 


72 


9-975974 


7 


! 54 


9-510434 


616 


•489566 '9-534504 


687 


10^465496 


-024070 


72 


9-975930 


6 


1 55 


9-510803 


615 


•489197 ! 9-534916 


687 


10-465084 


•024113 


72 


9-975887 


5 


1 56 


9-511172 


615 


•488828 !i 9-535328 


686 


10-464672 


•024156 


72 


9-975844 


4 


57 


9-511540 


614 


•488460 


9-535739 


686 


10-464261 


•024200 


72 


9-975800 


3 


1 58 


9-511907 


613 


•488093 


9^536150 


685 


10-463850 


-024243 


72 


9-975757 


2 


' 59 


9-512275 


613 


•487725 


9-536561 


685 


10-463439 


-024286 


72 


9-975714!! 1 


CO 


9-512642 612 


•487358 1 9-536972 


684 


10-463028 


•024330 


72 


9^975670 


j ' Cosiue. 


Secant. Cotai^^jpnt. 




Taugeut. 


Cosecant. 




Sine. 



71 DEG. 



60 



LOGARITHMIC SINES, ETC. 



19 


DE6. 






















Sine. 


Diff. 
100" 


Cosecant. 


1 Tangent. | f^,; 


Cotangent. 


Secant. 


Sn cosine, i; ' 1 





9-512642 




•487358 


9-536972 




10-463028 


! -024330 
1-024373 


|9-975670 60 


1 


9-513009 


612 


•486991 


9-537382 


684 


10-462618 


72 9-975627 59 


2 


9-513375 


611 


•486625 


9-537792 


683 


10-462208 


! -024417 


73 9-975583 :58 


3 


9-513741 


611 


•486259 


: 9-538202 


683 


10-461798 


-024461 


73!9-975539:57 


4 


9-514107 


610 


•485893 


9-538611 


682 


10-461389 


•024504 


73! 9 -97.5496:56 


5 


9-514472 


609 


■485528 


; 9-539020 


682 


10-460980 


-024548 


73 9-975452 ,55 


6 


9-514837 


609 


•485163 


9-539429 


681 


10-460571 


•024592 


73 9-975408 54 


7 


9-515202 


608 


•484798 


19-539837 


681 


10-460163 


•024635 


73 9-975365 53 


8 


9-515566 


608 


•484434 


9-540245 


680 


10-459755 


•024679 


73 9-975321 52 


9 jl 9-515930 


607 


•484070 


9-540653 


680 


10-459347 


-024723 


73 ,9-975277 51 


10 !i 9-516294 


607 


•483706 


; 9-541061 


679 


10-458939 


-024767 


73 9-975233 50 


11 9-516657 


606 


•483343 


19-541468 


679 


10-458532 


-024811 


73 9-975189 49 


12 ' 9-517020 


605 


•482980 


9-541875 


678 


10-458125 


1-024855 


73 9-975145 '48 


13 I 9-517382 


605 


•482618 


1 9-542281 


678 


10-457719 


i -024899 


73 '9-97510147 


14 9-517745 


604 


•482255 


9-542688 


677 


10-457312 


: -024943 


73 :9 -975057: 46 


15 :[ 9-518107 


604 


•481893 


19-543094 


677 


10-456906 


•024987 


73 :9-975013 !45 


16 19-518468 


603 


•481532 


9-543499 


676 


10-456501 


-025031 


73 


9-974969 


44 


17 19-518829 


603 


•481171 


9-543905 


676 


10-456095 


•025075 


74 19 -974925 


43 


181:9-519190 


602 


•480810 


9-544310 


675 


10-455690 


-025120! 74 19-974880 


42 


19 9-519551 


601 


•480449 


19-544715 


675 


10-455285 


'-025164! 74 19-974836 


41 


20 : 9-519911 


601 


•480089 


9-545119 


674 


10-454881 


•025208 


74 !9-974792 


40 


21 9-520271 


600 


•479729 


9-545524 


674 


10-454476 


-025252 


74 !9-974748 


39 


22 


9-520631 


600 


•479369 


9-545928 


673 


10-454072 


-025297 


74 |9-974703i!38 


28 


: 9-520990 


599 


•479010 


9-546331 


673 


10-4-53669 


-025341 


74 l9-974659| 37 
74 :9-974614i 36 


241:9-521349 


599 


•478651 


9-546735 


672 


10-453265 


-025386 


25 1 9-521707 


598 


•478293 


9-547138 


672 


10-452862 


•025430 


74 19 -9745701135 


26 j 9-522066 


598 


•477934 


9-547540 


671 


10-4-52460 


•025475 


74 9-974525:34 


271; 9-522424 


597 


•477576 


9-547943 


671 


10-452057 


•025519 


74 '9-97448i:;33 


281,9-522781 


596 


•477219 


9-548345 


670 


10-451655 


•025564 


74 19-974436} 32 


29 li 9-523138 


596 


•476862 


9-548747 


670 


10-451253 


•025609 


74 i9-974391ii31 


30 9-523495 


595 


•476505 


9-549149 


669 


10-4-50851 


•025653 


74 9-974347 30 


31 ii 9-523852 


595 


•476148 


9-549550 


669 


10-4-50450 ti-025698: 75 ,9-974302 


29 


32 |! 9-524208 


594 


•475792 


9^549951 


668 10-450049;: 0257431 75.9-974257 


,28 


33 1 9-524564- 594 


•475436 


9-550352 


668 10-449648 \ -025788 75 ;9-974212 


27 


34,19-524920: 593 


•475080 


9-550752 


667 10-449248 i -025833 75 ■9-974167 


26 


35! 9-525275' 593 


•474725 


9-551152 


667 110-448848 i -025878 75 9-974122 


;25 


36 9-525630' 592 


•474370 


9-551-552 


666 10-448448 '1-025923! 75 i9-974077 


24 


37 1 9-525984' 591 


•474016 


9-551952 


666 !l0-448048 


•025968; 75 19-974032 


23 


38; 9-526339:591 


•473661 


9-552351! 665 !l0-447649 


-026013! 75 9-973987 


;22 


39 1 9-526693 590 


•473307 


9^552750, 665 !l 0-447250 


-026058 75 19-973942 


21 


401,9-5270461590 


•472954 


9-553149, 665 |l-0-446851 


•026103 


75 9-973897 


I20 


41 i| 9-527400! 589 


•472600 


9-553548! 664110-446452 


•026148 


75 19-973852 


19 


42:^9-527753 589 


•472247 


9-553946! '664 |10-4460.54 


•026193 


75 9-973807 


18 


43' 9-528105; 588 


•471895 


9-554344 


663 ;10-445656 


-026239 


75i9-973761 


1.7 


44 9-528458; 588 


•471542 


9-554741 


663 10-445259 


-026284 


75 19-973716 


16 


45 19-528810 587 


•471190 


9-555139 


662 ilO-444861 


-026329 


76 9-973671 
76 19-973625 


15 


46 19-529161] 587^ 


•470839 


9-555536i 662 ilO-444464 


•026375 


14 


47 19-529513; 586 


•470487 


9-555933 


661 10-444067 


•026420 76'9^973580 


13 


48 1 9-529864 586 


•470136 


9-556329 


661 10-443671 


-026465' 76 9-973535 


12 


49; 9-530215; 585 


•469785 


9-556725 


660 !l0-443275 


•026511; 76 9^973489 


11 


50 '9-530565' 585 


•469435 


9-557121 


660 10^442879 


•026556! 76 9^973444 


10 


511 9-530915; 584 


•469085 


9^557517 


659 !l0-442483 


•026602! 76 j9^973398 


9 


52 9-531265' 584 


•468735 


9-557913 


659 |10-442087 


-026648' 76 9-973352 


8 


53 9-531614' 583 


•468386 


9-558308 


659 Il0-441692 


•026693 76 !9^973307| 


7 


54!! 9-531963' 582 


•468037 


9-558702 


658!] 0-441298 


•026739 


76 9^97326l| 


6 


55 i. 9-532312 582 


•467688 


19-559097 


658 ilO-440903 


-026785 


76 :9-973215| 


5 


56 [i 9-532661, 581 


•467339 


, 9-559491 


657 !l0-440509 


•026831 


76 9-9731691 


4 


57! 9-533009 581 


•466991 


9-559885 


657 |10-440115 


•026876 


76 19-9731241 


3 


58 !| 9-533357 580 


•466643 


9-560279 


656 !l0-439721 


•026922 


76 :9-973078' 


2 


59! 9-533704 580 


•466296 


9-560673 


656 10-439327 


•026968 


76 9-973032|i 1 1 


60 : 9-534052 579 


•465948 


9-561066 655:10-438934 


-027014! 76 9-972986 


-- 


1 ' Cosine. 


Secant. 


Cotangent.! Tangent. !' Cosecant. ' Sine. 


■ 1 



70 DEG. 



LOGARITHMIC SINES, ETC. 



61 



20 


DEG. 




















' 


8ine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


DifiF. 
100" 


Cosine. 


60 





9-534052 




•465948 


|9^661066 




10-438934 


•027014 




9-972986 


1 


9-534399 


578 


-465601 


1 9-561459 


655 


10-438541 


•027060 


77 


9-972940 


'59 


2 


9-534745 


577 


•465265 


19-561861 


654 


10-438149 


•027106 


77 


9-972894 


|58 


3 


9-535092 


577 


•464908 


9-562244 


654 


10-437756 


-027152 


77 


9-972848 


I57 


4 


9-535438 


577 


•464562 


9-562636 


653 


10-437364 


-027198 


77 


9-972802!!56 1 


5 


9-535783 


576 


•464217 


9-563028 


653 


10-436972 


•027246 


78 9-972756p56 


6 


9-536129 


576 


•463871 


9-563419 


653 


10-436681 


-027291 


77 9^972709^54 


7 


9-536474 


575 


•463526 


9-563811 


662 


10-436189 


-027337 


77 9^972663 


53 


8 


9-536818 


574 


-463182 


9-564202 


662 


10-435798 


-027383 


77 9^972617 


52 


9 


9-537163 


574 


•462837 


9-564592 


661 


10-435408 


-027430 


78 9^972570 


51 


10 


9-537507 


573 


-462493 


9-664983 


651 


10-436017 


-027476 


77 9^972524 


50 


111 


9-537851 


573 


•462149 


9-565373 


650 


10-434627 


-027622 


77 9^972478 


49 


121 


9-538194 


572 


•461806 


9-565763 


660 


10-434237 


•027569 


78 9-972431 


48 


13 


9-538538 


572 


•461462 


9-666153 


649 


10-433847 


-027615 


77 19-972385 


47 


14 


9-538880 


671 


-461120 


9-566542 


649 


10-433468 


-027662 


78 9-972338 


46 


15 


9-539223 


571 


•460777 


9-566932 


649 


10-433068 


-027709 


78 19-972291 


45 


16 


9-539565 


570 


-460435 


9-567320 


648 


10-432680 


-027755 


77 9-972245 


44 


17 


9-539907 


570 


•460093 


9-567709 


648 


10-432291 


-027802 


78 !9-972198 


43 


18 


9-540249 


569 


•459751 


9-568098 


647 


10-431902 


-027849 


78 9-972161 


42 


19 


9-540590 


669 


•459410 


9-568486 


647 


10-431514 


-027895 


77 I9-972105 


41 


20 


9-540931 


568 


•469069 


9-568873 


646 


10-431127 


-027942 


78 19-972058 


40 


21 


9-541272 


568 


•468728 


9-569261 


646 


10-430739 


•027989 


78 9-972011 


39 


22 


9-541613 


567 


•468387 


9-569648 


646 


10-430352 


-028036 


78 ;9-971964 


38 


23 


9-541953 


567 


•468047 


9-570035 


645 110-429965 1 


•028083 


78:9-971917 


37 


241 


9-542293 


566 


•467707 


9-570422 


646 


10-429578 


•028130 


78 19-971870 


36 


25 


9-542632 


566 


•457368 


9-670809 


644 


10-429191 


-028177 


78 19-971823 


j35 


26 


9-542971 


565 


•457029 


9-671195 


644 


10-428806 


•028224 


78 [9-971776 


34 


27! 


9-543310 


565 


-456690 


9-571581 


643 


10-428419 


-028271 


78 19-971729 


l33 


28 i 


9-543649 


564 


-456361 


9-571967 


643 


10-428033 


•028318 


78 '9-971682 


!32 


29! 


9-543987 


564 


-456013 


9-572362 


642 


10-427648 


-028366 


78 9-971635 


|31 


30 


9-544325 


563 


•455676 


9-572738 


642 


10-427262 


-028412 


78 9-971688 


30 


31 


9-544663 


563 


•465337 


9-573123 


642 


10-426877 


-028460 


80 19-971540 


29 


32 


9-545000 


562 


•466000 


9-673507 


641 


10-426493 


-028507 


78 


9-971493 


28 


33 1 


9-545338 


562 


•454662 


9-573892 


641 


10-426108 


-028554 


78 


9-971446 


27 


341 


9-545674 


561 


•464326 


9-674276 


640 


10-425724 


-028602 


80 


9-971398 


26 


35 


9-546011 


561 


-453989 


9-674660 


640 


10-425340 


-028649 


78 


9-971351 


25 


36 


9-546347 


660 


•453653 


9-676044 


639 


10-424966 


-028697 


80 


9-971303 


24 


37 


9-546683 


560 


•453317 


9-575427 


639 


10-424573 


-028744 


78 


9-971256 


23 


38 


9-547019 


569 


•452981 


9-576810 


639 


10-424190 


•028792 


80 


9-971208 


22 


39 


9-547354 


659 


•452646 


g^676193 


638 


10-423807 


•028839 


78 


9-971161 


21 


40 


9-547689 


558 


•452311 


9-576576 


638 


10-423424 


•028887 


80 


9-971113 


20 


41 


9-548024 


668 


•451976 


9-576958 


637 


10-423042 


•028934 


78 


9-971066 


19 


42 


9-548359 


557 


•451641 


9-577341 


637 


10-422659 


•028982 


80 


9-971018 


18 


43! 


9-648693 


657 


•451307 


9-577723 


636 


10-422277 


-029030 


80 


9^970970ll7 


44 


9-549027 


666 


•450973 


9-578104 


636 


10-421896 


-029078 


80 


9^970922 16 


45 


9-549360 


656 


-460640 


9-578486 


636 


10-421514 


-029126 


80 


9-970874 


15 


46 


9-549693 


665 


•450307 


9-578867 


635 


10-421133 


-029173 


78 


9-970827 


14 


47 


9-550026 


655 


-449974 


9-679248 


635 


10-420752 


-029221 


80 


9-970779 


13 


48 


9-550359 


554 


-449641 


9-679629 


634 


10-420371 


•029269 


80 


9-970731 


12 


49 


9-550692 


554 


•449308 


9-680009 


634 


10-419991 


•029317 


80 


9-970683 


11 


50 


9-551024 


563 


•448976 


9-580389 


634 


10-419611 


•029365 


80 


9-970635 


10 


51 


9-551356 


663 


•448644 


9-580769 


633 


10-419231 


•029414 


82 


9-970586 


9 


52 


9-551087 


552 


-448313 


9-581149 


633 


10-418851 


•029462 


80 


9-970538 


8 


53 


9-552018 


552 


•447982 


9-681528 


632 


10-418472 


•029510 


80 


9-970490 


7 


54 


9-552349 


552 


•447651 


9-581907 


632 


10-418093 


•029568 


80 


9-970442 


6 


55 


9-552680 


651 


•447320 


9-582286 


632 


10-417714 


•029606 


80 


9-970394 


5 


56 


9-553010 


561 


•446990 


9-582665 


631 


10-417335! 


•029665 


82 


9-970345 


4 


57 


9-553341 


650 


•446659 


9-583043 


631 


10-416957 ! 


•029703 


80 


9-970297 


3 


58 


9-553670 


550 


•446330 


9-583422 


630 


10-416578 


•029751 


80 


9-970249 


2 


59 


9-554000 


549 


•446000 


9-583800 


630 


10-416200 


•029800 


82 


9-970200 


1 


60 


9-554329 


649 


•445671 


9-684177 629 


10-415823 


•029848 


80 


9-970152 





' 


Cosine. 




Secant. 


1 Cotangent. 


Tangent. Cosecant. 




Sine. 


~^ 



69 DE«|. 



62 



LOGARITHMIC SINES, ETC. 



21 DEG. 





1 

2 
3 
4 

5i 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
341 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 i 
45 
46 i 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60! 
I — 



Sine. 



Diff. 
100" 



Cosecant. 



9-554329 

9-554658 
9-554987 
9-555315 
9-555643 
9-555971 
9-556299 
9-556626 
9-556953 
9-557280 
9-557606 
9-557982 
9-558258 
9-558583 
9-558909 
9-559234 
9-5.59.5-58 
9-5.59883 
9-560207 
9-560.531 
9-560855 
9-561178 
9-561501 
9-561824 
9-562146 
9-562468 
9-562790 
9-563112 
9-563433 
9-563755 
9-564075 
9-564396 
9-564716 
9-565036 
9-565356 
9-565676 
9-565995 
9-566314 
9-566632 
9-5669-51 
9-567269 
9-567587 
9-567904 
9-568222 
9-568539 
9-568856 
9-569172 
9-569488 
9-569804 
9-570120 
9-570435 
9-570751 
9-571066J 
9-571380' 
9-571695 
9-572009 
9-572323 
9-572636 
9-572950 
9-573263 
9-573575 



Cosine. 



-548 1 

548! 

547 1 

547 I 

546 

546 

545 

545 

544 

544 

543 

543 

543 

542 

542 

541 

.541 

540 

540 

539 

539 

538 

538 

537 

.537 

536 

536 

536 

535 

535! 

534 

534 

533 

533 

532 

532 

531 

531 

531 

530 I 

530 ! 

529 

529 

528 

528 I 

528 

527 I 

527! 

526 

526 I 

525 

525 j 

524 I 

524' 

523 

523 

523 i 

522 1 

522! 

521 ' 



•445671 
•445342 
•445013 
•444685 
•444357 
•444029 
•443701 
•443374 
•443047 
•442720 
•442394 
•442068 
•441742 
•441417 
•441091 
•440766 
•440442 
-440117 
•439793 
•439469 
•439145 
•438822 
•438499 
•438176 
•4378-54 
-437532 
•437210 
•436888 
•436567 
•436245 
•435925 
•435604 
•435284 
-434964 
•434644 
•434324 
•434005 
•433686 
•433368 
•433049 
•432731 
•432413 
•432096 
•431778 
•431461 
•431144 
•430828 
•430512 
•430196 
-429880 
-429565 
-429249 
•428934 
•428620 
•428305 
•427991 
•427677 
•427364 
•427050 
•426737 
•426425 



Tangent. 



Diff. 
100" 



Cotangent. 



9-584177 

9-584555 

9-584932 

9-585.309: 

9-585686! 

9-586062; 

9-586439 

9-5868151 

9 -.587 190 

9-587566 

9-587941 

9-588316 

9-588691 

9-589066 

9-589440 

9-589814 

9-590188 

9-590562, 

9-590935' 

9-5913081 

9-591681 

[9-592054^ 

! 9-592426' 

19 -5927 98: 

19-593171' 

! 9 -593.542 

J9-593914| 

j 9-594285! 

'9-594656' 

i9-595027j 

I 9-5953981 

I 9-5957681 

'9-596l38i 

: 9 •596.508: 

9-5968781 

; 9-597247: 

:9-597616' 

1 9-597985: 

I 9-598354' 

i 9-598722 

i 9-599091 

i 9-599459 

j 9-599827: 

i 9-600194 

' 9-600562! 

I 9-600929! 

19-601296! 

9-601662; 

9-602029: 

9-6023951 

. 9-602761J 

19-603127! 

i 9 -603493 

i 9-603858; 

i 9-6042231 

i 9-604588 

! 9-6049.53 

i 9-605317 

19-605682: 

i 9-6060461 

'9-606410! 



629 
629 
628 
628 
627 
627 
627 
626 
626 
625 
625 
625 
624 
624 
623 
623 
623 
622 
622 
622 
621 
621 
620 
620 
620 
619 
619 
618 
618 
618 
617 
617 
616 
616 
616 
615 
615 
615 
614 
614 
613 
613 
613 
612 
612 
611 
611 
611 
610 
610 
610 
609 
609 
609 
608 
608 
607 
607 
607 
606 



10-415823 

10-415445 

10-41.5068 

10-414691 

10-414814 

10-41.3938 

10-413.561 

10-413185 

10-412810 

10-412434 

10-412059 

10-411684 

10-411309 

10-410934 

10-410.560 

10-410186 

10-409812 

10-409438 

10-409065 

!10-408692 

!10-408819 

10-407946 

10-407574 

10-407202 

10-406829 

110-406458 

10-406086 

10-405715 

10-405844 

,10-404973 

40-404602 

I10-404232 

40-40.3862 

10-403492 

40-403122 

!l0-402753 

!l0-402384 

110-402015 

110-401646 

!10-401278 

ilO-400909 

110-400541 

10-400173 

:i0-399806 

;10-399438 

!10-399071 

|l0-398704 

10-398338 

10-397971 

|lO-397605 

ilO-3972.39 

110-396873 

10-396.507 !| 

10-396142 

10-395777 

10-39-5412 

10-395047 I 

10-394683 

10-394318 

10-393954 

10-393590 



Cotangent. 



Secant. 

029848 
029897' 
029945: 
029994 
030043 
030091! 
030140: 
0.30189 
030238 
030286, 
0.30335 
030384 
030433 
030482 
030531 
030.580 
030630 
030679' 
030728 
0307771 
0308271 
030876 
030925 
030975 
031024 
031074 
031123' 
031173 
081223 
031272^ 
031322! 
0318721 
031422! 
031472 
031521| 
0315711 
0316211 
031671 
0817221 
031772' 
031822i 
031872! 
031922 
031973 
032023 
032073 
032124 
032174 
032225 
032275 
032326! 
0-32876 1 
032427! 
0324781 
032529^ 
032579: 
032630' 
032681! 
032732 
0.32783 j 
032834 



Diff. 

'100" i 



Cos 



! 

i 



i9^970152 60 
81 9-970103 59 
81 9-9700-55 58 
81 9-970006 57 
81 '9-969957 56 
81 9-969909 55 
81 9-969860 54 
81 9-969811 53 
81 9-969762 52 
81 9-969714 51 
81 9-969665 50 

81 9-969616 49 

82 9-969567 ,48 
82 9-969518 47 
82 9-969469 46 
82 9-969420 45 
82 9-969370 44 
82 19-969321 43 
82 19-069272 42 
82 9-969223 41 
82 '9-969173 40 
82 i9^969124 39 
82 19-969075 38 
82 9-969025 37 
82 9^968976 36 

82 9-968926 35 

83 19-968877 34 
83 9-968827 33 
83 9-968777 32 
83 19-968728 31 
83 '9-968678 30 
83 i9-968628 29 
83 9-968578 28 
83 9-968528 27 
83 |9-968479 26 
83 |9-968429 25 
83 19-968379 24 
83 '9^968329 23 
83 9-968278 22 

83 ,9-968228 21 
84 '9-968178 ,20 

84 19-968128:19 
84 '9-968078 18 
84 i9-968027; 17 



84!9-967977| 
84 19-967927 
8419-967876! 
84 !9-967826! 
84 '9-967775 
84 '9^967725 
84 ^•967674 
84 '9^967624 
84 19-967573 

84 '9-967522 

85 19-967471 
85 '9 -967421 
85 19-967370 
85 19-967319 
85 9 -9672681 
85 19^967217 
85 9-967166! 



Cosecant. 



66 DJiG. 



LOGARITHMIC SINES, ETC. 



63 



22 


DEC 




ft 
















' 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. | ^^'^ 


Cotangent. 


Secant. 


Dift-. 
100" 


Cosine. 


' 





9-573576 




■ -426425 


1 9^606410 


10-393590 


-032834 




9-967166160 


1 


9-573888 


521 


•426112 


|9^606773|606 


10-393227 


•032885 


85 


9-967115:.59 


2 


9-574200 


520 


•425800 


9-6071371606 


10-392863 


-032936 


85 


9-967064 58 


3 


9-574512 


520 


•425488 


9-607500 605 


10-392500 


•032987 


85 


9-9670l3i;.57 


4 


9-574824 


519 


•425176 


9-607863' 605 


10-392137 


•033039 


85 


9-966961'56 


5 


9-575136 


519 


•424864 


9-608225; 604 


10-391775 


•033090 


85 


9-966910 55 


6 


9-575447 


519 


•424553 


1 9-608588; 604 


10-391412 


-033141 


85 


9-966859 54 


7 


9-575758 


518 


•424242 


9-608950i 604 


10-391050 


-033192 


85 


9-966808 53 


8 


9-576069 


518 


•423931 


19-609312 


603 


10-390688 


-033244 


85 


9-966756'!52 1 


9 


9-576379 


517 


•423621 


i 9-609674 


603 


10-390326 


-033295 


86 


9-96670551 i 


10 


9-576689 


517 


•423311 


i 9-610036 


603 


10-389964 


-033347 


86 


9-966653 50 | 


11 


9-576999 


516 


•423001 


19-610397 


602 


10-389603 


-033398 


86 


9-966602 


49 1 


12 


9-577309 


516 


•422691 


19-610759 


602 


10-389211 


-033450 


86 


9-966550; 


48 


13 


9-577618 


516 


•422382 


i 9-611120 


602 


10-388880 


-033501 


86 


9-966499 


47 


14 


9-577927 


515 


•422073 


9-611480 


601 


10-388520 


-033553 


86 


9-966447 


46 


15 


9-578236 


515 


•421764 


9-611841! 601 


10-388159 


•033605 


86 


9-966395' 


45 


16 


9-578545 514 


•421455 


1 9-612201! GOl 


10-387799 


-033656 


86 


9-966344 


44 


17 


9-578853 514 


•421147 


j 9-6125611 600 


10-387439 


•033708 


86 


9-966292 


43 


18 


9-579162 513 


•420838 


9-6129211 600 


10-387079 


•033760 


86 


9-966240 


42 


19 


9-579470 


513 


•420530 


9-613281, 600 


10-386719 


•033812 


86 


9-966188 


41 


20 


9-579777 


513 


•420223 


9-613641! .599 


10-386359 


-033864 


86 


9-96613640 1 


21 


9-580085 


512 


•419915 


9-614000 599 


10-386000 


-033915 


86 


9-966085 


39 


22 


9-580392 


512 


•419608 


9-6143-59; 598 


10-385641 


-033967 


87 


9-966033 


38 


23 


9-580699' 511 


•419301 


9-614718' 598 


10-385282 


-034019 


87 


9^965981 


37 


24 


9-5810051 511 


•418995 


9-61-5077 598 


10-384923 


-034071 


87 


9^965929 


36 


25 


9-581312 511 


•418688 


9-615435 


597 


10-384565 


-034124 


87 


9-965876 


35 


26 


9-581618 510 


•418382 


9-615793 


597 


10-384207 


•034176 


87 


9-965824 


34 


27 


9-581924 510 


•418076 


9-6161-51 


597 


10-383849 


-034228 


87 


9-965772 


33 


28 


9-582229; 509 


•417771 


9-616509 


596 


10-383491 


-034280 


87 


9-965720 


32 


29 


9-582535: 509 


•417465 


9-616867; 596 


10-383133 


-034332 


87 


9-965668 


31 


30 


9-582840; 509 


-417160 


9-617224! 596 


10-382776 


-034385 


87 


9-965615 


30 


31- 


9-583145' 508 


•416855|!9^617.582! 595 


10-382418 


•034437 


87 


9-965563 


29 


32 


9-583449: 508 


•416-551 j' 9-6179391 595 


10-382061 


•034489 


87 


9-965511 


28 


33 


9-583754 


507 


•4162461 9^618295! 595 


10-381705 


•034542 


87 


9-965458 


27 


34 


9-584058 


507 


-415942 9-6186-52 594 


10-381348 


•034594 


87 


9-965406 


26 


35 


9-584361 


506 


•415639 !i 9-619008^ 594 


10-380992 


•034647 


87 


9-965353 


25 


36 


9-584665 


506 


•415-335 'i 9-619364' 594 


10-380636 


-034699 


88 


9-965301 


24 


37 


9-584968 


506 


•415032 |i 9-6197211 593 


10-380279 


•034752 


88 


9-965248 


23 


38 


9-585272 


505 


-414728 1 19-620076; 593 


10-379924 


-034805 


88 


9-965195 


'22 


39 


9-585574 


505 


-414426 'I9-620432J -593 


10-379568 


-034857 


88 


9-965143 


21 


40 


9-585877 


504 


•414123 i 9-620787 592 


10-379213 


-034910 


88 


9-965090 


20 


41 


9-586179 


504 


•413821 1 9-621142 592 


10-378858 


-034963 


88 


9-965037 


19 


42 


9-586482 


503 


•413518 :9-621497i 592 


10-378503 


-035016 


88 


9-964984 


18 


43 


9-586783 


503 


•413217 , 9-621852 


591 


10-378148 


-035069 


88 


9-964931 


17 


44 


9-587085 


503 


-412915 i 9-622207 


591 


10-377793 


■035121 


88 


9-964879 


16 


45 


9-587386 


502 


•412614 1 9-622561 


590 


10-377439 


-035174 


88 


9-964826 


15 


46 


9-587688 


502 


•412312 li 9-622915 


590 


10-377085 


-035227 


88 


9-964773 


14 


47 


9-587989 


501 


•412011m 9-623269 


590 


10-376731 


-035280 


88 


9-964720 


13 


48 


9-588289 


501 


•411711 ij 9-623623 


589 


10-376377 


-035334 


h 


9-964666 


12 


49 


9-588590 


501 


-411410 i 9-623976 


589 


10-376024 


-035387 


89 


9-964613 


11 


50 


9-588890 


500 


-411110 9-624330 


589 


10-375670 


•035440 


89 


9-964560 


10 


51 


9-589190 


500 


•410810 .9-624683 


588 


10-375317 


-035493 


89 


9-964507; 


9 


52 


9-589489 


499 


•410511 9-625036 


588 


10-374964 


-035546 


89 


9-9644.54^ 


8 


53 


9-589789 


499 


•410211 9-625388 


588 


10-374612 


-035600 


89 


9-964400' 


7 


54 


9-590088 


499 


•409912;! 9^625741 


587 


10-374259 


•035653 


89 


9-964347 


6 


55 


9-590387 


498 


•409613' 9-626093 


587 


10-373907 


-035706 


89 


9-964294 


5 


56 


9-590686 


498 


•409314 |i 9^626445 


587 


10-373-555 


-035760 


89 


9-964240 


4 


57 


9-590984 


497 


•409016 1 9-626797 


586 


10-373203 


-035813 


89 


9-9641871 


3 


58 


9-591282 


497 


•408718 9-627149 
•408420 9-627501 


586 


10-372851 


-035867 


89 


9-964133' 


2 


59 


9-591580 


497 


586 


10-372499 


-035920 


89 


9-964080; 


1 


60 


9-591878 


496 


•408122 9 •627852 


585 


10-372148 


•035974 


89 


9-964026 





Cosine. 




Secant. 1 Cotangent. 




Tangent. 


Cosecant. 




Sine. 


"^ 



67 DEG. 



64 



LOGARITHMIC SINES, ETC. 



23 


DEG. 























Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


f^r, Cotangent. | 


Secant. 


Diff. 
100" 


Cosine. 


60 


9-591878 




•408122 


9-627852 


10-3721481; 


•035974| 


9^964026: 


1 


9-592176 


496 


•407824 


9-628203 


585 10-3717971 


-036028; 89 


9-968972'59 


2 


9-592473 


495 


•407527 


9-628554 


585 


10-371446 ; 


-036081! 89 


9-963919!58 


3 


9-592770 


495 


•407230 


9-628905 


585 


10-371095 ' 


-086135: 89 


9-963865157 


4 


9-593067 


495 


•406933 


9-629255 


584 


10-370745 


-086189: 90 


9-963811 '56 


5 


9-593363 


494 


•406637 


9-629606 


584 


10-370394 


-086243 90 


9-963757,55 


6 


9-593659 


494 


-406341 


9-629956 


583 


10-370044 


-036296 90 


9-963704!54 


7 


9-598955 


493 


•406045 


9-630306 


583 


10-369694 


-036850 90 


9-9636501.53 


8 


9-594251 


493 


•405749 


9-630656 


583 110-369344 


-036404 90 '9-968596 52 i 


9 


9-594547 


493 


•405453 


9-631005 


588 10-368995 


-036458! 90 


9-963542:51 


10 


9-594842 


492 


•405158 


9-631355 


582! 10-868645 


-036512! 90 


9-963488 50 


11 


9-595137 


492 


•404863 


9-631704 


582jl0-368296 


-036566; 90 


9-963434 49 j 


12 


9-595432 


491 


•404568 


9-632053 


582 10-367947 


-036621190 


9-968879 48 


13 


9-595727 


491 


•404273 


9-632401 


581 10-367599 


•0366761 90 


9-963325 47 


14 


9-596021 


491 


•403979 


9-6327.50 


581 ilO-867250 


-036729! 90 


9-96.3271 46 


16 


9-596315 


490 


•403685 


9-633098 


581110-366902 


-086783! 90 


9-963217 45 


16 


9-596609 


490 


•403391 


9-688447 


580 


10-366558 ! 


-036837! 90 


9-963163 44 


17 


9-596903 


489 


•403097 


9-688795 


580 


10«66205' 


•0368921 90 


9-963108 43 


18 


9-597196 


489 


•402804 


9-684148 


580 


10-365857 1 


-036946J91 
-03700lf91 


9-968054 42 


19 


9-597490 


489 


•402510 


9-684490 


579 


10-865510 j 


9-962999 41 


20 


9-597783 


488 


•402217 


9-684838 


579 


10-365162! 


-037055' 91 


9-962945 40 


21 


9-598075 


488 


•401925 


9-685185 


579 


10-364815 


-037110! 91 


9-962890 39 


22 


9-598368 


487 


•401632 


9-685532 


578 


10-364468 


-087164 91 


9-962836 38 


23 


9-598660 


487 


.401340 


9-635879 


578 


10-364121 i 


-087219 91 


9-962781 37 


24 


9-598952 


487 


•401048 


9-636226 


578 


10-363774 1 


-037273 91 


9-962727 36 


25 


9-599244 


486 


•400756 


9-636572 


577 


10-863428 


-037328 91 


9-96267235 


26 


9-599536 


486 


•400464 


9-636919 


577 


10-363081 1 


•037883 91 


9-962617|34 


27 


9-599827 


485 


•400173 


9-637265 


577 


10-362735 


-087488! 91 


9-962562'33 


28 


9-600118 


485 


•399882 


9-637611 


577 


10-362389 ' 


-037492 91 


9-962508 32 


29 


9-600409 


485 


•399591 


9-687956 


576 


10-.^62044. 


-037547 91 


9-962458! 31 


30 


9-600700 


484 


•399300 


9-638802 


576 


10-361698 


-037602 91 


9-962398 30 


31 


9-600990 


484 


•399010 


9-638647 


576 


10-861853 


-037657! 92 


9-962848 29 


32 


9-601280 


484 


•898720 


9-638992 


575 


10-361008 


-037712; 92 


9-962288 28 


33 


9601570 


483 


•398430 


9-639337 


575 


10-860663 


-087767 92 


9-962233 27 


34 


9-601860 


483 


•398140 


9-639682 


575 


10-360318 


-087822 92 


9-96217826 


35 


9-602150 


482 


•397850 


9-640027 


574 


10-3.59973 : 


-0878771 92 


9-962123,25 


36 


9-602439 


482 


•397561 


9-640371 


574 10-359629 I 


-087988 92 


9 -962067 24 


37 


9-602728 


482 


•397272 


9-640716 


574 10-359284' 


-037988 92 


9-962012 23 


38 


9-603017 


481 


•396983 


9-641060 


573 10-358940! 


-038048 


92 


9-961957 22 


39 


9-603305 


481 


.396695 


9-641404 


573 10-358596 ! 


-038098 


92 


9-961902 21 


40 


9-603594 


481 


•896406 


9-641747 


578 110-858253 


•0381.54 


92 


9-961846 20 


41 


9-603882 


480 


•396118 


9-642091 


572 10-857909! 


•038209 


92 


9-96179l!|19 


42 


9-604170 


480 


.395830 


9-642434 


572|lO-357.566! 


-038265 


92 


9-961735118 


43 


9-604457 


479 


•395543 


9-642777 


572 10-357*223 1 


-038320 


92 


9-961680 17 ; 


44 


9-604745 


479 


•395255 


9-643120 


572 ilO-356880 ' 


-038376 


92 


9-961624 16 | 


45 


9-605032 


479 


•894968 


9-643463 


571 il0-3o6587 : 


-038431 


98 


9-961569 15 j 


46 


9-605319 


478 


.394681 


9-643806 


571 


10-356194 


•038487 


93 


9-961513 14 j 


47 


9-605606 


478 


•394394 


9-644148 


571 


10-355852 


•038542 


93 


9-961458 13 | 


48 


9-605892 


478 


•394108 


9-644490 


570 


10-35.5510 


-038598 


93 


9-961402 12 i 


49 


9-606179 


477 


•393821 


9-644832 


570 


10-355168 


-038654 


93 


9-961846 11 i 


50 


9-606465 


477 


.393535 


9-645174 


570 


10-354826 : 


-088710 


93 


9-961290 10 1 


51 


9-606751 


476 


.393249 9-645516 


570 


10-354484 ■ 


-038765 


93 


9-961235, 9 , 


52 


9-607036 


476 


.392964! 9-645857 


569 


10-354143 j 
10-358801 ! 


-088821 


93 


9-9611791 8 i 


53 


9-607322 


476 


.392678 9-646199 


569 


-088877 


93 


9-961123 i 7 1 


54 


9-607607 


475 


•392393 


9-646540 


569 


10-353460 .; 


-088933 


93 


9-961067il 6 


55 


9-607892 


475 


•392108 


'. 9-646881 


568 


10-353119 |: 


-088989 


93 


9-9610111 5 


56 


9-608177 


474 


.891823 ' 9-647222 


568 


10-352778!! 


-039045 


93 


9-960955 1 4 


57 


9-608461 


474 


.3915391 9-647562 


568110-352438,! 


-039101 


93 


9-960899 ; 3 1 


58 


' 9-608745 


474 


•391255, ,9-647903 


567 


10-352097 


-039157 


93 


9-960843 1 2 


59 


9-609029 


473 


.390971 


9-648243 


667 


10-351757 


•039214 


94 


9-960786 j 1 


60 


9-609313 


473 


.390687 


j 9-648583 567 


10-351417 


•039270 


94 


9-960730! 1 


1 ' 


1 Cosine. 




Secant. || Cotangent. | 


Tangent. 


Cosecant. 




Sine. ,i ' 1 



66 DEG. 



LOGARITHMIC SINES, ETC. 



65 



24 DEG. 



1 


Sine. 


Di«f. 

1()0" 


Cosecant. 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


iDiff 
100' 


Cosine. 


, 





9-609313 




-390687 


9^648583 




10-351417 


•039270 




9-960730 


, 

60 


1 


9-609597 


473 


•390403 


9-648923 


566 


10-351077 


•039326! 94 


9-960674 


59 


2 


9-609880 


472 


•390120 


! 9^649263 


566 


10-350737 


•039382 94 


9-960618 i58 


3 


9-610164 


472 


•389836 


9^649602 


566 


10-350398 


•0394391 94 


9-960561 i57 


4 


9-610447 


472 


•589553 


9-649942 


566 


10^350058 


•039495 i 94 


9-960505 '56 


■5 


9-610729 


471 


-389271 


9-650281 


565 


10^349719 


•039552 


94 


9-960448i!55 


6 


9-611012 


471 


•388988 


9^650620 


565 


10^349380 


•039608 


94 


9-960392!i54 


7 


9-611294 


470 


•388706 


9^650959 


565 


10^349041 


•039665 


94 


9-960335 


153 


8 


9-611576 


470 


•388424 


9-651297 


564 


10-348703 


•039721 


94 


9-960279 


52 


9 


9-611858 


470 


•388142 


9-651636 


564 


10^348364 


•039778 


94 


9-960222 


51 


10 


9-612140 


469 


•387860 


9-651974 


564 


10^348026 


•039835 


94 


9-960165 


50 


11 


9-612421 


469 


•387579 


9-652312 


563 


10-347688 


•039891 


94 


9-960109 


!49 


12 


9-612702 


469 


•387298 


9-652650 


563 


10-347350 


•039948 


95 


9-960052 


48 


13 


9-612983 


468 


•387017 


9-652988 


563 


10-347012 


•040005 


95 


9-959995 


47 


14 


9-613264 


468 


•386736 


9-653326 


563 


10-346674 


•040062 


95 


9-959938 


46 


15 


9-613545 


467 


•386455 


9-653663 


562 


10-346337 


-040118 


95 


9-959882 


45 


16 


9-613825 


467 


•386175 


9-054000 


562 


10-346000 


-040175 


95 


9-959825 


44 


17 ! 9-614105 


467 


•385895 


9-654337 


562 


10-34-5663 


-040232 


95 


9-959768 


43 


18 p 9-614885 


466 


•385615 


9-654674 


561 


10-345326 


•040289 


95 


9^959711 


42 


19 ! 9-614665 


466 


•385335 


9-655011 


561 


10-344989 


•040346 


95 


9^959654 


41 


201:9-614944 


466 


-385056 


9-655348 


561 


10-344652 


•040404 


95 


9^959596 


!40 


21 i 9-615223 


465 


-384777 


9-655684 


561 


10-344316 


•040461 


95 


9^959539;:39 


22 9-615502 


465 


-384498 


9-656020 


560 


10-343980 


•040518 


95 


9-959482;!38 


23 19-615781 


465 


-384219 


9-656356 


560 


10-343644 


-040575 


95 


9-959425137 


24 i! 9-616060 


464 


-383940 


9-656692 


560 


10-343308 


-040632 


95 


9-959368:|36 


25: 9-616338 


464 


-383662 


9-657028 


559 


10-342972 


•040690 


95 


9-959310:!35 


26 9-616616 


464 


•383384 


9-657364 


559 


10-342636 


•040747 


96 


9-959253'^34 


27 '9-616894 


463 


•383106 


9-657699 


559 


10-342301 


•040805 


96 


9-959195{33 


28 9-617172 


463 


•382828 


9-658034 


5-59 


10-341966 


•040862 


96 


9^959138''32 


29 9-617450 


462 


•382550 


9-658369 


558 


10-341631 


•040919 


96 


9^959080 :31 


30 19-617727 


462 


•382273 


9-658704 


558 


10-341296 


•040977 


96 


9^959023 j30 


31 ; 9-618004 


462 


•381996 


9-659039 


558 


10-340961 


•041035 


96 


9^958965 29 


32 i 9-618281 


461 


•381719 


9-659373 


558 


10-340627 


•041092 


96 


9^958908!28 


33 i: 9-618558 461 


•381442 


9-659708 


557 


10-340292 


•041150 


96 


9^958850: 27 


34 


9-618834 461 


•381166 


9-660042 


557 


10-339958 


•041208 


96 


9-958792126 


35 


9-619110 460 


•380890 


9-660376 


557 


10-339624 


•041266 


96 


9-958734 25 


36 


9-619386^ 460 


•380614 


9-660710 


556 


10-339290 


•041323 


96 


9-958677 24 


37'! 9-619662 460 


•380338 


9-661043 


556 


10-338957 


•041381 


96 


9-958619 


23 


38 9-619938! 459 


•380062 


9-661377 


556 


10-338623 


•041439 


96 


9-958561 


22 


39:^9-620213 4-59 


•379787 


9-661710 


556 


10-338290 


•041497 


96 


9-958503 


21 


40 jj 9-620488: 459 


•379512 


9-662043 


555 


10-337957 


•041555 


97 


9-958445 


20 


41 '19-620763 458 


•379237 


9-662376 


555 


10-337624 


•041613 


97 


9-958387 


19 


42 li 9-621038 458 


•378962 


9-662709 


555 


10-337291 


•041671 


97 


9-958329 


18 


43 


9-621313 457 


•378687 


9-663042 


554 


10-336958 


•041729 


97 


9-958271 


17 


44 


9-621587 457 


•378413 


9-663o75 554 


10-336625 


•041787 


97 


9-958213 


16 


45 


9-621861 457 


-378139 


9-663707 554 


10-336293 


•041846 


97 


9-958154 


15 


46 


9-622135J 456 


-377865 


9-664039 


554 


10-33-5961 


•041904 


97 


9-958096 


14 


47 


9-622409 456 


•377591 


9-664371 


553 


10-335629 


-041962 


97 9-958038 


13 


48 


9-622682' 456 


•377318 


9-664703 


553 


10-335297 


-042021 


97 [9-957979 


12 


49 


9-622956 455 


-377044 


9-665035 


553 


10-334965 


-042079 


97 ;9-957921 


11 


50 


j 9-623229 455 


-376771 


9-665366 


553 


10-334634 


-042137 


97!9-957863! 


10 


51 


! 9-623502 455 


-376498 


9-665697 


552 


10^334303 


•042196 


97 


9-957804 


9 


52 i 9-623774 454 


-376226 


9-666029 


552 


10-333971 


•042254 


97 


9-957746J 


8 


53 19-624047 454 


•375953 


9-666360 


552 


10-333640 


•042313 98 


9-957687| 


7 


54 9-624319 454 


•375681 


9-666691 


551 


10-333309 


•042372 


98 


9-9576281 


6 


55 i 9-624591 453 


•375409 


9-667021 


551 


10-332979 


•042430 


98 


9-9575701 


5 


56 ! 9-624863 453 


•375137 


9-667352 


551 


10-332648 


•042489 


98 


9-9575111 


4 


57;! 9-625135 453 


•374865 


9-667682 


551 


10-332318 


•042548 


98 


9-957452 


3 


58 |! 9-625406 452 


•374594 


9-668013 


550 


10-331987 


•042607 


98 


9-957393! 


2 


59 '19-625677 452 


•374323 


9-668343 


550 


10-331657 


•042665 


98 


9-9573351 


1 


60 9-625948 452 


•374052 


9-668672 


550 


10^331328 


•042724 


98 


9-957276 





' , Cosine. 


Secant. 


Cotangent. 




Tangent. ! 


Cosecant. 




Sine. 1 



65 DUG. 



m 



26 


DEG. 




















' i 


Sine. 


Diff. 

100" 


Cosecant. j| Tangent. 


^ 1 Cotangent. 


Secant. 


Diff. 

100" 


j 1 

Cosine. , ' j 


^\ 


9-625948 




-374052 i 19-668673 


'10-331327 


•042724 




9-957276^60 


li 


9-626219 


451 


-373781 ' 9-669002 


550 10-330998 


•042783 


98 


9-957217' 69 


2i 


9-626490 


451 


•373510 i! 9-669332 


549 !l0-330668 


-042842 


98 


9-957158 58 1 


el 


9-626760 


451 


•373240 19-669661 


549 10-330339 


•042901 


98 


9-957099 57 I 


4 


9-627030 


450 


•372970 19-669991 


549 10-330009 


-042960 


98 


9-957040 56 


5 


9-627300 


450 


•372700 ; 9-670320 


548 10-329680 


-043019 


98 


9-956981 :66 | 


6| 


9-627570 


450 


•372430, 9-670649 


548 ,10-329351 


-043079 


98 


9-956921 '54 ! 


7| 


9-627840 


449 


•372160 '9^670977 


548 10-329023 


•043138 


99 


9-956862 63 


8| 


9-628109 


449 


•371891; 9^671306 


648 i 10-328694 


•043197 


99 


9-956803 52 


9i 


9-628348 


449 


•371622! 9^671634 


547 !l0-328366 


•043256 


99|9-956744.51 


10 


9-628647 


448 


•3713531 9-671963 


547 ilO-328037 


•043316 


9919-956684 ;50 


111 


9-628916 


448 


•371084!: 9-672291 


647 ;iO-327709 


•043376 


99|9-956626,49 i 


12 i 


9-629185 


447 


•370815; 9-672619 


547 '10-327381 


•043434 


99 9-956566 '48 ; 


13; 


9-629453 


447 


•370547;; 9-672947 


546 10-327053 


•043494 


99 9-956506147 1 


14 il 9-629721 


447 


-370279;; 9-673274 


546 !l0-326726 


•043553 


99 


9-966447'l46 


15i 


9-629989 


446 


-370011 ; 9-673602 


546 |l0-326398 


-043613 


99 


9-956387^ 45 1 


16! 


9-630257 


446 


•369743;! 9^673929 


546 110-326071 


-043673 


99 


9-956327i44 i 


17' 


9-630524 


446 


•369476!! 9-674257 


545 ilO-325743 


-043732 


99 


9-966268!43 | 


18! 


9-630792 


446 


-369208: 9-674584 


545 |10-325416 


-043792 


99 


9-966208;;42 1 


19 


9-631059 


445 


•368941; 9-674910 


545 1 10-325090 


-043852 


100 


9-956148 


'41 


20 


9-631326 


445 


•368674 9-675237 


544 ;i0-324763 


•043911 


100 


9 •956089 


|40 


21 il 9-631593 


445 


•368407 ; 9-675564 


544 


10-324436 


-043971 


100 


9^956029 


39 


22 1 


9-631859 


444 


-368141 


9^675890 


644 


10-324110 


-044031 


100 


9-955969 


38 


23 


9-632125 


444 


-367875 


9-676217 


544 


10-323783 


-044091 


100 


9-955909 


37 


24 j 


9-632392 


444 


-367608 


9-676543 


543 !l0-323457 


-044161 


100!9-955849!!36 


25' 


9-632658 


443 


•367342 


9-676869 


643 110-323131 


-044211 


100 9-955789135 


261 


9-632923 


443 


•367077 19-677194 


543 !l0-322806 


-044271 


100|9-956729|l34 


271 


9-633189 


443 


•366811 1 9-677520 


643 |l0-322480 


-044331 


100 9-955669 33 


28; 


9-633454 


442 


-366546 : 9-677846 


542 |l0-322154 


•044391 


100i9-955609''32 


29! 


9-633719 


442 


-366281 1 9-678170 


642 ilO-321829 


•044452 


100 9-955548 31 


301 


.9-633984 


442 


•366016; 9-678496 


542 110-321604 


-044512 100 9-956488 30 j 


31 !! 9-634249 


441 


•365751!; 9-678821 


542|l0-321179 


•044672;100 9-955428 129 | 


32; 9-634514 


441 


-365486 Ij 9-679146 


641 10-320854 


-044632 


101 


9-955368 28 j 


i 33 1 9-634778 


440 


•365222 ; 9-679471 


541 10-320529 


-044693 


101 


9-955307 27 \ 


34; 9-635042 
35 j 9-635306 


440 


-36495819-679795 


541 10-320205 


-044753 


101 


9-955247 26 


440 


-364694;! 9-680120 


541 


10-319880 


-044814 


101 


9-955186 25 


36 


9-635570 


439 


-364430! 9-680444 


540 


10-319566 


•044874 


101 


9-955126 24 


37 


9-635834 


439 


•364166!' 9-680768 


540 


10-319232 


-044936 


101 


9-965065 23 j 


38 


9-636097 


439 


-363903 9-681092 


640 


10-318908 


-044995 


101 


9-955005 22 i 


39 


9-636360 


438 


•363640 9-681416 


640 


10-318584 


-045056 


101 


9-954944 21 1 


40 


9-636623 


438 


•363377^9-681740 


539 


10-318260 


-045117 


101 


9-954883 20 ! 


41 


9-636886 


438 


•363114 9-682063 


639 


10-317937 


•045177 


101 


9-954823 19 | 


42 i! 9-637148 


437 


-362852; 9-682387 


539 


10-317613 


•046238' 101 


9-954762 18 


43!; 9-637411 


437 


•362589 9-682710 


539 110-317290 


•045299:101 


9-954701 17 


44 ij 9-637673 


437 


•362327 19-683033 


538 


10-316967 


-046360101 


9-954640 16 


45' 9-637935 


437 


•362065,1 9^683356 


638 


10-316644 


-045421 jlOl 


9-954579 15 


46 


19-638197 


436 


-361803;! 9-683679 


538 


10-316321 


-045482101 


9-954518114 


47 


9-638458 


436 


•361542!, 9-684001 


538 


10-315999 


-046543 102 


9-954467:113 


48 


9-638720 


436 


-361280 '9-684324 


637 


10-316676 


•045604 102 


9-9543961112 


49 


9-638981 


435 


■361019 1 9-684646 


537 


10-315354 


•045666 102 


9-954335;!ll 


50 


9-639242 


435 


-360758 ! 9-684968 


537 


10-315032 


•0457261102 


9-954274i!l0 1 


51 :| 9-639503 


435 


•360497 i 9-685290 


537 


10-314710 


•045787[102 


9-954213!| 9 


52119-639764 


434 


-360236 ! 9-685612 


636 


10-314388 


•045848 102 


9-954l52i! 8 ! 


53 ii 9-640024 


434 


•359976 ; 9-685934 


636 


10-314066 


•0459101102 


9-954090l| 7 


54 !j 9-640284 


434 


•359716 i 9^686255 


536 


10-313746 


•045971! 102 


9-954029; 6 | 


55 i 9-640544 


433 


•359456 : 9-686577 


536 


10313423 


•046032!l02 


9-953968! 5 


56; 9-640804 


433 


-359196,! 9-6868981 535 


10-313102 


•046094|102 


9-953906! 4 


1 67 'j 9-641064 


433 


•358936 


9-687219 535 


10-312781 


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9-953845! 3 


58: 9-641324 


432 


•358676 


9-687540 535 


10-312460 


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9-963783' 2 ! 


59 19-641583 


432 


•358417 


9-687861 535 


10-312139 


-046278 102l9-953722!l 1 ' 


60 


j 9-641842 


432 


•358158 


9-688182; 534 10-311818 


•046340 103!9^953660i i 

! ; li 1 


i 1_ 


! Cosine. 


1 Socant. 


i Cotangent, i Tangent. Cosecant. ] Sine. !| ' | 



64 DEG. 



LOGARITHMIC SINES, ETC. 



67 



26 


DEC. 




















' 


Sine. 


Diff. 

100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


^ cosine, i; ' 1 





9-641842 




-358158 


9-688182 




10-311818 


-04634t) 


9-953660|J00 | 
103 9-953599159 | 


1 


9-642101 


431 


-357899 


9-688502 


634 


10-311498 


-046401 


2 


9-642360 


431 


-357640 


9-688823 


534 


10-311177 


-046463 


103 9-953537 


68 1 


3 


9-642618 


431 


-357382 


9-689143 


534 


10-310867 


-046526 


103 9-963475 


57 


4 


9-642877 


430 


-357123 


9-689463 


533 


10-310637 


-0465871103 9-9534131 


56 


5 


9-643135 


430 


-356865 


9-689783 


533 


10-310217 


-0466481103 9-953352':56 1 


6 


9-643393| 430 


-356607 


9-690103 


533 


10-309897 


-046710 


103 9-953290:54 i 


7 


9-643650| 430 


-356350 


9-690423 


533 


10-309577 


-046772 


103 9-953228.163 


8 


9-643908; 429 


-366092 


9-690742 


533 


10-309258 


-046834 


103 9-953166':52 


9 


9-6441651 429 


-365835 


9-691062 


532 


10-308938 


•046896 


103 9-963104!:61 


10 


9-6444231 429 


-355577 


9-691381 


632 


10-308619 


•046958 


103;9-963042: 60 i 


11 


9-644680! 428 


-355320 


9-691700 


532 


10-308300 


-047020 


103 9-952980 49 


12 


9-644936| 428 


-366064 


9-092019 


631 


10-307981 


-047082 


104 9-962918:|48 1 


13 


9-6451931 428 


-354807 


9-692338 


531 


10-307662 


-047146 


104 9-962856' 


47 


14 


9-6454501 427 


-364560 


9-692656 


531 


10-307344 


-047207 


104 9-962793' 


46 


15 


9-645706 427 


•354294 


9-692975 


531 


10-307025 


-047269 


104 9-9627311 


45 


15! 


9-64o962i 427 


-854038 


9-693293 


631 


10-306707 


-047331 


1049-962669i 


44 


I'l 


9-646218 


426 


-353782 


9-693612 


530 


10-306388 


-047394 


1049-962606;43 


18 


9-646474 


426 


•353626 


9-693930 


530 


10-306070 


-047466 


104 9-952544i;42 


19 


9-646729 


426 


-363271 


9-694248 


530 


10-305752 


-047619 


104 9-962481i'41 


20 1 


9-646984 


425 


-363016 


9-694666 


530 


10-305434 


-047681 


104 9-962419 '40 


21 


9-647240 


425 


-352760 


9-694883 


529 


10-306117 


-047644J104 9-952356 '39 | 


22 


9-647494 


425 


-352506 


9-695201 


529 


10-304799 


-047706 


104 9-952294; 38 


23 


9-647749 


424 


-362261 


9-696618 


529 


10-304482 


-047769 


104^9-962231: 37 


2-1; 


9-648004 


424 


-361996 


9-696836 


629 


10-304164 


-047832 


104'9-952108'36 


25 


9-648258 


424 


-361742 


9-696163 


529 


10-303847 


-047894 


105,9-962106' 36 


26 


9-648512 


424 


-361488 


9-6964701 528 


10-303630 


-047967 


105 9-962043,34 


27 


9-648766 


423 


•351234 


9-696787 528 


10-303213 


-048020 


105 9-951980 33 


.28 


9-649020 


423 


•350980 


9-697103] 528 


10-302897 


•048083 


105 9-951917 32 


29 


9-649274 


423 


•360726 


9-697420 528 


10-302680 


•048146 


105 9-951854 131 


30 


9-649527 


422 


•350473 


9-697736 


527 


10-302264 


•048209 


105|9-96179li'30 


31 


9-649781 


422 


•350219 


9-698053 


527 


10-301947 


•048272 


105|9-961728| 


29 


32 


9-650034 


422 


•349966 


9-698369 


527 


10-301631 


•048336 


105,9-9616651 


28 


33 


9-650287 


422 


•349713! 


9-698685 


527 


10-301315 


•048398 


105 


9-951602 


27 


34 


9-650539 


421 


•349461 1 


9-699001 


526 


10-300999 


-048461 


105 


9-961639 


26 


35 


9-650792 


421 


•349208 ' 


9-699316 


526 


10-300684 


-048524 


106 


9-951476 


25 


36 


9-651044 


421 


•348956 


9-699632 


526 


10-300368 


-048588 


105 


9-951412 


24 


37 


9-651297 


420 


•348703 


9-699947 


526 


10-300063 


•048651 


105 


9-951349 


23 


38 


9-651549 


420 


•348451 


9-700263 


526 


10-299737 


•048714 


106 


9-951286 


22 


39 


9-651800 


420 


•348200 


9-700578 


525 


10-299422 


-048778 


106 


9-961222 


21 


40 


9-652052 


419 


•347948 


9-700893 


525 


10-299107 


-048841 


106 


9-951159 


20 


41 


9-652304 


419 


•347696 


9-701208 


525 


10-298792 


-048904 


106 


9-951096 


19 


42 


9-652555 


419 


•347445 


9-701523 


525 


10-298477 


-048968 


106 


9-961032 


18 


43 


9-652806 


418 


•347194 


9-701837 


524 


10-298163 


-049032 


106 


9-950968 


17 


44 


9-653057 


418 


•346943 


9-702152 


524 


10-297848 


-049096 


106 


9-960905|ll6 
9-960841115 


45 


9-653308 


418 


•346692 


9-702466 


524 


10-297534 


-049159 


106 


46 


9-653558 


418 


•346442 


9-702780 


524 


10-297220 


•049222 


106 


9-960778 


|14 


47 


9-653808 


417 


•346192 


9^703095 


523 


10-296905 


•049280 


106 


9-960714 


13 


48 


9-654059 


417 


•345941 


9-703409 


523 


10-296691 


•049350 


106 


9-950650 


12 


49 


9-654309 


417 


•345691 


9-703723 


523 


10-296277 


-049414 


106 


9-950586,|ll 1 


50 


9-654558 


416 


•345442 


' 9-704036 


523 


10-295964 


•049478|l06i9-960522 


10 


51 


9-654808 


416 


-345192 


' 9-704350 


523 


10-295650 


•049542 107i9-950458 


9 


52 


9-655058 


416 


-344942 


9-704663 


522 


10-295337 


•049606 107i9-950394 


8 


53 


9-655307 


415 


-344693 


9-704977 


522 


10-295023 


•049670|107!9^950330 


7 


54 


9-655556 


415 


-344444 


9-706290 


522 


10-294710 


-049734 


10719-950266 


6 


55 


9-655805 


415 


-344195 


' 9-705603 


522 


10-294397 


-049798 


10719-950202 


5 


56 


9-656054 


415 


•343946 


1 9-705916 


521 


10-294084 


•049862 


107i9-950138 


4 


57 


9-656302 


414 


-343698 


9-706228 


621 


10-293772 


•049926 


10719-950074 


3 


58 


9-666551 


414 


-343449 


9-706641 


521 


10-293459 


•049990|107!9-950010 


2 


59 


9-656799 


414 


-343201 


9-706854 


521 


10-293146 


-060056 


107 9^949945 


1 


60 


9-657047 


413 


-342953 9-707166 


521 


10-292834 


-050119 


107 9-949881 





Cosine. 


1 Secant. Cotangent. | 


Tangent. 


Cosecant. 


Sine. 


'^ 



36 



08 



LOGARITHMIC SINES, ETC. 



27 


PEG, 




















, 1 


• 
Sine. 


foO^- 1 Cosecant. 


Tangent. | fj^; 


Cotangent. 


Secant. 

•050119 


Diff. 
100" 


Cosine. 


1 
6O" 





9-657047 




342953 


9-7071661 


10-292834 


9-949881 


1 


9-657295 


413 


342705 


9-707478: 520 ^10-292522 i!-050184 


107 


9^949816 


59 


9 


9-G5754-J 


413 


34245^ 


9-707790 520 '10-292210 i-050248 


107 


9-949752 


58 


3 


9-657790 


412! 


342210 


9-708102; 520 ,10-291898 :i-050312 107;9-949688 


57 


4 


9-G5y037 


412 


3419i33 


9-708414, 520 10-291586 :l-0-50377 108 9-949623 


56 


61 


9-658284 


412 


341716 


9-708726; 519 10-291274 !|-0.50442;i08:9-9495.58 


56 


6| 


9-658531 


412 


341469 


9-709037: 519 110-290963 !j-0.50506ll08!9-949494 


;54 


7 1 


9-658778 411 | 


341222 


9-709349 519 10-290651 |!-05057l!l08:9-949429 


i53 


8 


9-659025 


411 


340975 


9-709660: 519 10-290340 ^0.50636a08'9-949364"52 i 


9 


9-659271 


411 


340729 


9-7099711 519 10-290029 j!-0-50700;i08:9-949300 '51 ! 
9-710282! 518 :10-289718i!-050765!l08 9-949235 '50 i 


10 i 


9-659517 


410 


340483 


11 


9-659763 


410 


340237 


9-710.593! 518!l0-289407 


-0508-30 108 9-949170 49 


12 


9-660U09 


410 


339991 


9-710904, 518 


10-289096 


•050895!l08 9-949105; 48 


13 


9-660255 


409 


339745 


9-711215 518 


10-288785 


-050960!l08;9-949040;47 


14 


9-660501 


409 


339499 


9-711525} 518 


10-288475 


-051025 108,9-948975' 46 


15 


9-660746 


409 


3392-54 


9-711836' 517 


10-288164 


•051090|l089-948910l45 


16 


9-660991 


409 


339009 


9-712146 517 


10-287854 


•051155 108 9-948845 !44 


17 


9-661236 


408 


338764 


9-712456 517 


10-287544 


-051220il08;9-948780;i43 


18 


9-661481 


408 


338519 


9-712766; 517 


10-287234 


-051285 


109 9-948715;;42 


19! 


9-661726 


408 


338274 


9-713076 516 


10-286924 


•051350 


109,9-948650:41 


20 j 


9-661970 


407 


338030 


9-7133861 516 


10-286614 


•051416 


109 9-948584':40 


21 


9-662214 


407 


337786 


9-713696| 516 


10-286304 


-051481 


109|9-948519;39 


22 


9-6624-59 


407 


337541 


9-714005 


516 


10-285995 


-051546 


109 9.9484541S38 


23 


9-662708 


407 


337297 


9-714314 


516 


10-285686 


•051612J109!9-948388 '37 | 


24! 


9-662946 


406 


337054 


9-714624 


515 


10-285376 


•051677 


109 


9-948323;;36 


251 


9-663190 


406 


336810 


9-714933 


515 


10-285067 


•051743 


109 


9-948257135 


261 


9-663433 


406 


336567 


9-715242 


515 


10-284758 


-051808 


109 


9-948192!'34 


27 1 


9-663677 


405 


336323 


9-7155511 515 


10-284449 


-051874 


109 


9^948126!33 


28 


9-663920 


405 


336080 


9-7158601 514 


10-284140 


•051940 


109 


9^948060 32 


29! 


9-664163 


405 


335837 


9-716168! 514 


10-283832 


-052005 


109 


9^947995! 31 


30 j 


9-664406 


405 


335594 


9-716477! 514 


10-283523 


-0-52071 


110!9^947929 


30 


31 


9-664648 


404 


335352 


9-716785 514 


10-283215 


•0-52137 


110!9^947863 


29 


32 1 


9-664891 


404 


335109 


9-717093; 514 


10-282907 


-052203 


110'9^947797 


28 


33 i 


9-665133 


404 


334867 


9-717401! 513 


10-282599 


-0-52269 


1109-947731 


|27 


34! 


9-665375 


403 


334625 


9-717709! 513 !l0-282291 


-052335 


110!9-947665 


26 


35! 


9-665617 


403 


334383 


9-718017; 513 !l0-281983 


•052400 


110'9-947600 


!25 


36 


9-665859 


403 


334141 


9-718325! 513 !l0-281675 


-052467 


110:9-947533 


124 


37 


9-666100 


402 


333900 


9-718633' 513 !l0-281367 


•0-52533!ll0!9-947467 


23 


38 


9-666342 


402 


333658 


9-718940! 512 !l0-281060 


•052599 110!9-947401 


22 


39 


9-666583 


402 


333417 


9-719248^ 512 {10-280752 


•052665 IIOI9-947335 


21 


40 


9-666824 


402 


333176 


9-719555 512 10-280445 


•052731 110!9-947269 


20 


41 


9-667065 


401 


332935 


9-719862 512 ilO-280138 


-0-52797 110;9-947203 


19 


42 


9-667305 


401 


332695 


9-720169 512 !l0-279831 


-052864 110 9-947136 


18 


43 


9-667546 


401 


332454 


9-7204761 511 jlO-279524 


-052930;ill!9-947070 


17 


44 


9-667786 


401 


-332214 


9-720783 


511 10-279217 


-052996;ili;9-947004 


16 


45 


9-668027 


400 


331973 


9-721089 


511 ilO-278911 


•0-53063!lli:9-946937 


15 


46 


9-668267 


400 


331733 


9-721396 


511 110-278604 


•053129 


111:9-946871 


14 


47 


9-668506 


400 


331494 


9-721702 


511 ilO-278298 


•053196 


lll|9-946804 


13 


48 


9-668746 


399 


3312-54 


9-722009 


510 10-277991 


-053262 


1119-946738 


12 


49 


9-668986 


399 


331014 


9-722315 


510 !l0-277685 


-053329'111'9-946671 


11 


50 


9-669225 


399 


330775 


9-722621 


510 10-277379 


-0.53396!lll 9-946604 


10 


51 


9-669464 


399 


330536 


9-722927 


510 10-277073 


-053462 


111 


9^946538 


9 


52 


9-669703 


398 


-330297 


9-723232 


510 10-276768 


-053529 


111 


9-946471 


8 


53 


9-669942 


398 


330058 


9-723538 


509 :i0-276462 


•053596 


111 


9-946404 


7 


54 


9-670181 


398 


329819 


9-723844 


509 


10-276156 


•053663!lll 


9-946337 


6 


55 


9-670419 


397 


329581 


9-724149 


509 


10-275851 


•053730;ill 


9-946270 


5 


56 


9-670658 


397 


329342 


9-724454 


609 10-275-546 


•053797lll2!9-946203 


4 


57 


9-670896 


397 


329104 


9-724759 


609 110-275241 


-053864!112:9-946136 


3 


58 


9-671134 


397 


328866 


9-725065 


508 10-274935 


•05393l!ll2 


9-946069 


2 


69 


9-671372 


396 


328628 


9-725369 508110-274631 


•053998,112 


9-946002 


1 


60 




9-671609 


396 


328391 


9-725674 


-508! 10-274326 


•054065 112 


9-945935 





Cosine. 


Sec-ant. •' Cotangent. 


i Tangent. 


Cosecant. 


Sine. 



62 DEG. 



28 


DEG. 


• 


dj\f\j[ 


mvxxxxiuj. 


V> VJJL. 


iiiaoj XJJLV 








Uc' 


' 


Sine. 


Diff. 

100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. Secant. 


Diff. 

100" 


Cosine, j ' 





9-671609 




•328391 


9-725674 




10-274326 


•054065 




9-946935 60 


1 


9-671847 


396 


•328163 


9-725979 


508 


10-274021 


•064132 


112 


9-945868' 59 


2 


9-672084 


395 


•327916 


9-726284 


508 


10-273716 


•054200 


112 


9-94.5800 58 


3 


9-672321 


396 


•327679 


9-726588 


607 


10-273412 


-054267 


112 


9 -94.57331 57 


4 


9-672558 


395 


•327442 


9-726892 


507 


10-273108 


-054334 


112 


9-945666 56 


6 


9-672795 


895 


•327206 


9^727197 


507 


10-272803 


-064402 


112 


9-945598!l55 


6 


9-673032 


394 


•326968 


9-727501 


607 


10-272499 


-054469 


112 


9-94553li!54 


7 


9-673268 


394 


•326732 


9-727805 


507 


10-272195 
10-271891 


•054536 


112 


9-945464; 


53 


8 


9-673505 


394 


•326495 


9-728109 


606 


-064604 


113 


9-945396 


52 


9 


9-673741 


394 


•326259 


9-728412 


506 


10-271.588 


-054672 


113 


9-945328 


61 


10 


9-673977 


393 


•326023 


9-728716 


506 


10-271284 


-054739 


113 


9-945261 


50 


11 


9-674213 


393 


•326787 


9-729020 


606 


10-270980 


•064807 


113 


9-945193 


49 


12 


9-674448 


393 


•326562 


9-729323 


506 


10-270677 


•054875 


113 


9-945125 


48 


13 


9-674684 


392 


•326316 


9-729626 


605 


10-270374 


•054942 


113 


9-945068 


47 


14 


9-674919 


392 


•325081 


9-729929 


505 


10-270071 


-056010 


113 


9-944990 


46 


15 


9-675165 


392 


•324846 


9-730233 


506 


10-269767 


•055078 


113 


9-944922 


45 


10 


9-675390 


392 


•324610 


9-73063O 


605 


10-269465 


•066146 


113 


9^944854 


44 


17 


9-675624 


391 


•324376 


9-730838 


505 


10-269162 


-056214 


113 


9^944786 


43 


18 


9-676859 


391 


•324141 


9-731141 


504 


10-268869 


-066282 


113 


9-9447^8 


42 


19 


9-676094 


391 


•323906 


9-731444 


604 


10-268556 


-056360 


113 


9-944650 


41 


20 


9-676328 


391 


•323672 


9-731746 


504 


10-268254 


-055418 


113 


9-944582 


40 


21 


9-676562 


390 


•323438 


9-732048 


504 


10-267952 


•066486 


114 


9-944514 


39 


22 


9-676796 


390 


•323204 


9-732351 


604 


10-267649 


•055564 


114 


9-944446 


38 


23 


9-677030 


390 


•322970 


9^732663 


503 


10-267347 


-055623 


114 


9-944377 


37 


24 


9-677264 


390 


•322736 


9-732955 


503 


10-267045 


•056691 


114 


9-944309 


36 


25 


9-677498 


389 


•322602 


9-733257 


603 


10-266743 


•055759 


114 


9-944241 


36 


26 


9-677731 


389 


•322269 


9-733658 


603 


10-266442 


•055828 


114 


9-944172 


34 


27 


9-677964 


389 


•322036 


9-733860 


603 


10-266140 


•056896 


114 


9-944104 


33 


28 


9-678197 


388 


•321803 


9-734162 


503 


10-265838 


-056964 


114 


9-944036 


32 


29 


9-678430 


388 


•321570 


9-734463 


502 


10-266.537 


•066033 


114 


9-943967 


31 


30 


9-678663 


388 


•321337 


9-734764 


502 


10-265236 


•066102 


114 


9-943899 


30 


31 


9-678895 


388 


•321105 


9-736066 


602 


10^264934 


•056170 


114 


9-943830 


29 


32 


9-679128 


387 


•320872 


9-735367 


502 


10^264633 


•056239 


114 


9-943761 


28 


33 


9-679360 


387 


•320.640 


9-736668 


502 


10-264332 


•056307 


114 


9-943693 


27 


34 


9-679692 


387 


•320408 


9-736969 


601 


10-264031 


•056376 


115 


9-943624 


26 i 


35 


9-679824 


387 


•320176 


9-736269 


501 


10-263731 


•056445 


115 


9-943565 


25 


36 


9-680056 


386 


•319944 


9-736670 


501 


10-263430 


•056514 


116 


9-943486 


24 


37 


9-680288 


386 


•319712 


9-736871 


601 


10-263129 


•056583 


115 


9-943417 


23 


38 


9-680519 


386 


•319481 


9-737171 


501 


10-262829 


•056652 


115 


9-943348 


22 


39 


9-680760 


385 


•3192.50 


9-737471 


500 


10-262629 


•056721 


115 


9-943279 


21 


40 


9-680982 


385 


•319018 


9-737771 


500 


10-262229 


•056790 


115 


9-943210 


20 


41 


9-681213 


385 


•318787 


9-738071 


500 


10-261929 


•056869 


116 


9-943141 


19 


42 


9-681443 


385 


•318557 


9-738371 


500 


10-261629 


•066928 


115 


9-943072 


18 


43 


9-681674 


384 


•318326 


9-738671 


500 


10-261329 


•056997 


115 


9-943003 


17 


44 


9-681906 


384 


•318096 


9-738971 


500 


10-261029 


•057066 


116 


9-942934 


16 


45 


9-682136 


384 


•317865 


9-739271 


499 


10-260729 


•057136 


115 


9-942864 


15 


46 


9-682366 


384 


•317635 


9-739670 


499 


10-260430 


•057205 


116 


9-942795 


14 


47 


9-682595 


383 


•317405 


9-739870 
1 9-740169 


499 


10-260130 


•057274 


116 


9-942726 


13 


48 


9-682825 


383 


•317176 


499 


10-259831 


•057344 


116 


9-942656 


12 


49 


9-683056 


383 


•316945 


19-740468 


499 


10-269632 


•057413 


116 


9-942587 


11 


50 


9-683284 


383 


•316716 


9-740767 


498 


10-2.59233 


-057483 


116 


9-942517 


10 


51 


9-683614 


382 


•316486 


9-741066 


498 


10-268934 


•057552 


116 


9-942448 


9 


62 


9-683743 


382 


•316257 


9-741365 


498 


10-258635 


•067622 


116 


9-942378 


8 


53 


9-683972 


382 


•316028 


9^741664 


498 


10-268336 


•057692 


116 


9-942308 


7 


54 


9-684201 


382 


•315799 


; 9-741962 


498 


10-258038 


•057761 


116 


9-942239 


6 


55 


9-684430 


381 


•315670 


j 9-742261 


498 


10-257739 


•057831 


116 


9-942169 


5 


56 


9-684658 


381 


•315342 


I 9-742569 


497 


10-257441 


•057901 


116 


9-942099 


4 


57 


9-684887 


381 


•315113 


i 9-742858 


497 


10-257142 


•057971 


116 


9-942029 


3 


58 


9-685115 


380 


•314885 


19-743156 


497 


10-256844 


•058041 


116 


9-941959 


2 


59 


9-685343 


380 


•314667 


19-743454 


497 


10-256546 


-058111 


116 


9-941889 


1 


60 


9-686671 


380 


•314429 


19-743752 


497 


10-256248 


-058181 


117 


9-941819 





' 


1 Cosiue. 


! Secant, l! Cotangent. 




Tangent. 


Cosecant. 




Sine. 1 


~^ 



61 DE(J. 



70 



LOGARITHMIC SINES, ETC. 



29 


DEG. 




















1 ' 


Sine. 


Diff. 
100" 


Cosecant. 


j Tangent. 


DiflF. 
100" 


Cotangent. 


Secant. 


Diff 
100" 


Cosine. 







; 9-685571 




•314429 


! 9-743752 




10-256248 


•058181 




9-941819 


60 


! 1 


; 9-685799 


380! 


314201 


i 9-744050 


496 


10-256950 


•068261 


117 


9-941749 


69 


1 2 


! 9-686027 


379: 


313973 


1 9-744348 


496 


10-256652 


•058321 


117 


|9-941679 


58 


3 ii 9-686264 


379; 


3137461; 9-744645 


496 


10-256355 


•068391 


117 


9-941609 


57 


4 ;i 9-686482 


379 


3135181' 9-744943 


496 


10-256057 


•058461 


117j9^941539 


56 


5 li 9-686709 


379 


313291 il 9-745^40 


496 


10-254760 


•058631 


1179-941469 


155 


6 :; 9-686936 


378 i 


313064 1 9-745538 


496 


10-254462 


•058602 


1179-941398 


54 


7!! 9-687163 


378' 


312837 ! 9-745835 


495 


10-254165 


•058672 


1179-941328 


53 


8119-687389 


378: 


3126111! 9-746132 


495 


10-253868 


•058742 


117t9-941258 


52 


9 |i 9-687616 


378! 


312384^9-746429 


495 


10-253571 


•068813 


11719-941187 


61 


10 


! 9-687843 


377 i 


312157 19-746726 


495 


10-253274 


•068883 


117'9-941117 


|50 


11 


19-688069 


377 


311931 i| 9-747023 


495 


10-262977 


•058954 


117 9-941046!|49 i 


12 


' 9-688295 


377 


311705^19-747319 


494 


10-252681 


•0590261118 9-940975;i48 | 


13 


19-688521 


377 


311479 


9-747616 


494 


10-252384 


•059095 


118 9-940905 


147 


14 


1 9-688747 


376 


311253 


9-747913 


494 


10-252087 


•059166 


118:9-940834 


46 


15 


] 9-688972 


376 


311028 


9-748209 


494 


10-251791 


•059237 


118:9-940763 


45 


16 li 9-689198 


376 


310802 19-748505 


494 


10-251495 


-059367 


11819-940693 


44 


17 li 9-689423 


376 


310577 11 9-748801 


494 


10-251199 


•059378 


118 


9-940622 


43 


18 |i 9-689648 


375 


310352 Ij 9-749097 


493 


10-250903 


•059449 


118 


9^940551 


42 


19119-689873 


375 


310127,19-749393 


493 


10-250607 


•059520 


118 


9^940480 


41 


20 i! 9-690098 


375 


309902 


9-749689 


493 


10-250311 


•059591 


118 


9-940409 


40 


21} 9-690323 


375 


309677 . 


9-749985 


493 


10-250015 


-059662 


118 


9-940338 


39 


221:9-690548 


374 


309452 


9-750281 


493 


10-249719 


•059733 


118 


9-940267 


38 


23 i 9-690772 


374 


309228 


9-750576 


493 


10-249424 


•059804 


118 


9-940196 


37 


24 i| 9-690996 


374 


309004 II 9-750872 


492 


10-249128 


•069875 


118 


9^940125 


36 


25 


19-691220 


374 


30878019-751167 


492 


10-248833 


•059946 


119 


9-940054 


35 


26 


9-691444 


373 


308556 jl 9-751462 


492 


10-248538 


-060018 


119 


9-939982 


34 


27 


19-691668 


373 


308332:19-751757 


492 


10-248243 


-060089 


119 


9-939911 


33 


28 li 9-691892 


373 


308108 i 9-752052 


492 


10-247948 


-060160 


119 


9-939840 


32 


29 jl 9-692115 


373 


30788511 9-752347 


491 


10-247653 


•060232 


119 


9-939768 


31 


30 II 9-692339 


372 


307661 19-7-52642 


491 


10-247358 


•060303 


119!9-939697 


30 


1 311! 9-692562 


372 


307438 


9-752937 


491 


10-247063 


•060375 


119 


9-939625 


29 


1 32 i; 9-692785 


372 


307215 


9-753231 


491 


10-246769 


-060446 


119 


9-939-554 


28 


33 19-693008 371 


306992 


9-753526 


491 


10-246474 


•060518 


119 


9-939482 


27 


34 l! 9-693231 371 


306769 


9-753820 


491 


10-246180 


-060590 


119 


9-939410 


26 


i 351:9-693453 371 


306547 


9-754115 


490 


10-245885 


•060661 


119 


9-939339 


25 


36 J! 9-693676 371 


306324 


9-754409 


490 


10-246691 


-060733 


119 


9-939267 


24 


3711 9-693898 370 


306102 


9-754703 


490 


10-245297 


•060805 


120 


9-939196 


23 


38 II 9-694120 370 


305880 


9-754997 


490 


10-245003 


•060877 


120 


9-939123 


22 


i 39 11 9-69434-2; 370 


305658 


9-755291 


490 


10-244709 


-060948 


120 


9-939052 


21 


1 40 II 9-694564 370 


305436 


9-755585 


490 


10-244415 


-061020 


120 


9-938980 


20 


Ul i' 9-694786 369 


305214 


9-755878 


489 


10-244122 


•061092 


120 


9-938908 


19 


i 42 1 9-695007! 369 


304993 


9-756172 


489 


10-243828 


•061164 


120 


9-938836 


18 


1 43 [19-695229 369 


304771 


9-756465 


489 


10-243535 


•061237 


120 


9-938763! 


17 


1 44 i 9-695450 369 


304550 


9-756759 


489 


10-243241 


•061309 


120 


9-938691 


16 


45 19-6956711368 


304329 


9-757052 


489 


10-242948 


•061381 


120 


9-938619 


15 


46 19-695892 368 


304108 


9-757345 


489 


10-242665 


-061453 


120 


9-938547 


14 


47 19-696113: 368 


303887 


9-757638 


488 


10-242362 


-061526 


120 


9-^38475 


13 


48 119-696334 368 


303666 


9-757931 


488 


10-242069 


-061598 


120 


9-938402 


12 


j 49119-696554 367 


303446 


9-758224 


488 


10-241776 


•061670 


121 


9^938330 


11 


\ 50 119-696775 367 


303225 


9-768517 


488 


10-241483 


•061742 


121 


9^938258 


10 


511; 9-696995 367 


303005 


9-758810 


488 


10-241190 


-061815 


121 


9^938185 


9 


1 52 ;i 9-697215 367 


302785 


9-759102 


488 


10-240898 


•061887 


121 


9-938113 


8 


1 58119-697435 366 


302565 


9-759395 


487 


10-240605 


•061960 


121 


9-938040 


7 


64 i| 9-697654 366 


302346 


9-759687 


487 


10-240313 


•062033 


121 


9-937967 


6 


65 11 9-697874 366 


302126 


9-759979 


487 


10-240021 


•062105 


121 


9-937895 


6 1 


56 II 9-698094 366 


301906 


9-760272 


487 


10-239728 


•062178 


121 


9-937822 


4 


1 57119-698313 365 


301687 


9-760564 


487 


10-239436 


-062251 


121 


9^937749 


3 


1 58 j| 9-698532 365 


301468 


9-760856 


487 


10-239144 


•062324 


121 


9^937676 


2 


59 b 9-698751 365 


301249 


9-761148 


486 


10-238852 


•062396 


121 


9^937604 


1 


601' 9-698970 365 


301030 


9-761439 


486 


10-238561 


•062469 


121 


9^937531 





1 ' Cosine. 


Secant. 


Cotangent. 




Tangent. 


Cosecant. 




Sine. 



60 psa. 



LOGARITHMIC SINES, ETC. 



30 


DEC. 






















Sine. 


DiS. 
100" 


Cosecant, i | Tangent. 


f^; 1 Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


' 





9-698970 




•301030 9 


761439 


il0^238561 


•062469 




9-937531 


eT 


1 


9-699189 


364 


•300811 9 


761731 


486 10^238269 


•062542 


121 


9-937458 59 


2 


9-699407 


364 


-300593 9 


762023 


486 !l0^237977 


•062615 


122 


9^937385 58 


3 


9-699626 


364 


-300374 9 


762314 


486 !10^237686 


•062688 


122 


9-937312157 


4 


9-699844 


364 


•300156 9 


762606 


486 10-237394 


•062762 


122 


9^937238li56 


5 


9-700062 


363 


•299938 9 


762897 


485 10-237103 


•062835 


122 


9-937165 55 


6 


9-700280 


363 


•299520 9 


763188 


485 1 10-236812 


-062908 


122 


9 •9370921 ^54 


! 7; 


9-700498 363 


•299702 9 


763479 


485 10-236521 


-062981 


122 


9-937019 53 


8 


9-700716 


363 


•299284 9 


763770 


485 10-236230 


-063054 


122 


9-9369461152 


9 


9-700933 


363 


•299067 9 


764061 


485 110-235939 


•063128 


122 


9-936872, 51 


10 1 


9-701151 


362 


•298849 : 9 


764352 


485 10-235648 


•063201 


122 


9^936799 '50 


1 Hi 


9-701368 


362 


•298632 9 


764643 


485 10-235357 


-063275 


122 


9-936725 149 


1 121 


9-701585 


362 


•298415 9 


764933 


484 10-235067 


■063348 


122 


9-936652 '48 


13 


9-701802 


362 


•298198 9 


765224 


484il0-234776 


•063422 


123 


9-936678';47 


14 


9-702019 


361 


•297981 9 


765514 


484 


10-234486 


•063495 


123 


9-936505'46 


15! 


9-702236 


361 


•297764 9 


765805 


484 


10-234195 


•063569 


123 


9-936431'45 


16' 


9-702452 


361 


•297548 9 


766095 


484 


10-233905 


•063643 123 


9-936357;44 


I'i 


9-702669 


361 


•297331 9 


766385 


484 


10-233615 


•063716 


123 


9-936284|!43 


18 


9-702885 


360 


•297115 9 


766675 


483 


10-233325 


•063790 


123 


9-936210142 


19 i 


9-703101 


360 


•296899, 9 


766965 


483 


10-233035 


-063864 


123 


9-936136 


41 


20 i 


9-703317 


360 


•296683 9 


767255 


483 


10^2S2745 


-0639381123 


9-936062 


ko 


21! 


9-703533 


360 


•296467 9 


767545 


483 


10^232455 


•064012 123 


9-935988 


39 1 


22! 


9-703749 


359 


•296251 9 


767834 


483 


10^232166 


•064086 123 


9-935914! 38 


231 


9-703964 


359 


•296036 9 


768124 


483 


10-231876 


•064160123 


9-935840 37 


24 


9-704179 


359 


•295821 9 


768414 


482 


10-231586 


•004234123 


9-935766!36 


25 j 


9-704395 


359 


•295605 9 


768703 


482 


10-231297 


•064308124 


9 -935692! :35 


26 i 


9-704610 359 


•295390 9 


768992 


482 


10-231008 


•0643821124 


9-935618134 


27 


9-704825 


358 


•295175 9 


769281 


482J10-230719 


•064457124 


9-935543' 33 


28 j 


9-705040 


358 


•294960 9 


769570 


482 10^230430 


•064531124 


9-935469:32 


29 


9-705254 


358 


•294746 9 


769860 


482 10-230140 


•0646051 124 


9-935395131 


1 30j 


9-705469 


358 


•294531 9 


770148 


481 i 10^229852 


•0646801124 


9-935320 30 


31! 


9-705683 


357 


•294317 9 


770437 481 


10-229563 


•064754 


124 


9^935246i;29 


32 


9-705898 


357 


•294102 1 9 


7707261 481 


10-229274 


•064829 


124 


9^93517l||28 


' 33 1 


9-706112 


357 


•293888 9 


771015i 481 


10-228985 


•064903 


124 


9^935097'i27 


i 34! 


9-706326 


357 


•293674 9 


771303: 481 


10-228697 


•064978 


124 


9^93502226 


36 


9-706539 


356 


•293461 9 


771592 481 


10-228408 


•065052 


124 


9-934948 25 


36 j 


9-706753 


356 


•293247 1 9 


7718S0i 481 


10-228120 


•065127 


124 


9^934873:24 


37 


9-706967 


356 


•293033 9 


772168: 480 


10-227832 


•065202 


124 


9^934798 123 


38: 


9-707J80 


356 


•292820 9 


7724571 480 


10-227543 


•065277 


125 


9^934723 22 


39. 


9-707393 


355 


•292607 9 


772745; 480 


10-227255 


•065351 125 


9^934649 21 


40, 


9-707606 


355 


•292394 {! 9 


773033 480 


10-226967 


•005426125 


9^984574 20 


41 


9-707819 


355 


•292181 9 


773321 480il0-226679i 


•065501125 


9^934499 19 


42; 


9-708032 


355 


•291968 9 


773608 480 


10-226392 


•065576 125 


9^934424 il8 


43 1 


9-708245 


354 


•29175511 9 


773896 480 


10-226104 


•065651 125 


9^934349 17 


i 44 


9-708458 


354 


•291542 I! 9 


7741841 479 


10-225816 


•065726125 


9^934274 16 


1 45 


9-708670 


354 


•291330 9 


774471^479 


10-225529 


•065801125 


9-934199:15 I 


! 46 


9-708882 


354 


•291118 9 


774759, 479 


10-225241 


•065877 125 


9-934123 14 


! 47 


9-709094 


353 


•290906 t 9 


775046' 479 


10-224954 


•065952 125 


9^934048 13 


48 


9-709306 


353 


•290694:; 9 


775333 479 


10^224667 


•066027 125 


9^933973 12 


49 


9-709518 


353 


•290482:19 


775621 479 10-224379 


•066102 125 


9^933898 11 


50 


9-709730 


353 


•2902701! 9 


775908 478 ll0-224092 


•066178 126 


9^933822: 10 


51 


9-709941 


353 


•290059 9 


776195 478il0-223805i 


•066253 126 


9-933747:, 9 


52 


9-710153 


352 


•289847 !! 9 


776482 478 10-223518' 


•066329 126 


9-933671'! 8 


53 


9-710364 


352 


•289636 9 


776769 478 ,10-223231 ; 


•066404 126 


9^933596:; 7 


54 


9-710575 


352 


•289425 ii 9 


777055 478110-222945! 


•066480 126 


9 •933520' 6 


55 


9-710786 


352 


•289214 j 9 


777342 478110-222658: 


•066555 126' 


9-933445'i 5 


56 


9-710997 


351 


•289003 i 9 


777628 47810-222372: 


•066631 126 


9^933369 4 


57 


9-711208 


351 


•288792 9 


777915 477 10-222085 -066707 126, 


9^933293: 3 


! 58 


9-711419 351 


•288581 9 


778201 477 10-221799 |-066783 126 


9 •93321 7 2 


59 


9-711629 


351 


•288371 9 


778487 477 10^221ol3 :-066859 126, 


9 -933 141; 1 


60 


9-711839 


350 


•288161 9^778774 477 j 10-221226 •066.»34 126 


9-933066 i 




; Cosini-. 


acc.'iiu. Cotuugtnt. T;iug.-ia. C-.-caul. , .Muu. ' \ 



59 DEG. 



72 



LOGARITHMIC SINES, ETC. 



31 DEG. 



Tangent. ^^; I Cotangent. Secant. ^^;, Cosine. 

! 



Sine. 





1 

. 2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

1 12 

i 13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 i 

27 i 
28^ 



^fj^; Cosecant. 



29 9 

30 9 



31 



33 I 

34; 

35 1 9 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



711839 
712050 
712260 
■712469 
712679 
712889 
713098 
713308 
713517 
713726 
713935 
714144 
•7143521 
•7145611 
•714769! 
•714978i 
•715186J 
•715394, 
•715602^ 
•715809! 
•7160171 
■7162241 
•7164321 
■7166391 
•7168461 
■717053 
•7172591 
•717466! 
•717673 
■717879! 
•7180851 
•718291^ 
•7184971 
•718703 
•718909 
•719114 
•719320 
•719525 
719730 
719935 
720140 
720345 
720549 
720754 
•720958 
•723162 
■721366 
■721570 
■721774 
•721978 
•722181 
•722385 
•722588 
•722791 
•722994 
•723197 
■723400 
■723603 
■7^3805 
724007 
■724210 



Cosine. 



350 

350 

350 

349 

349 

349 

349 

349 

348 

348 

348 

348 

347 

347 

347 

347 

347 

346 

346 

346 

346 

345 

345 

345 

345 

345 

344 

344 

344 

344 

343 

343 

343 

343' 

343' 

342 1 

342 1 

342 I 

3421 

341 i 

341 I 

3411 

341 i 

340' 

340 

340 

340 

340 I 

3391 

339 i 

339, 

339: 

339! 



337 
337 
337 



•288161 

•287950 

•287740 

■287531 

•287321 

•287111 

•286902 

•286692 

•286483 

•286274 

•286065 

•285856 |! 9 

•285648 11 9 

•285439 1 1 9 

•285231 1! 9 

•285022 

-.284814 

•284606 

•284398 

•284191 

•283983 

•283776 

•283568 

•283361 

•283154 

•282947 

•282741 

•282534 

•282327 

•282121 

•281915 

•281709 

•281503 

•281297 

•281091 



•280680 
•280475 
•280270 
•280065 
•279860 
•279655 
•279451 
•279246 
•279042 
•278838 
•278634 
•278430 
•278226 
•278022 
■277819 
•277615 
•277412 
•277209 
•277006 
•276803 
•276600 
•276397 
•276195 
•275993 
•275790 



778774 
779060 
779346 
779632 
779918 
780203 
780489 
780775 
781060 
781346 
781631 
781916 
782201 
•782486 
782771 
783056 
783341 
783626! 4 
783910 
784195 
784479! 4 
784764 
785048 
785332 
•785616 
•785900 
•786184 
•786468 
•786752 
•787036 
•787319 
•787603 
•787886 
•788170 4 
•788453 
■788736 
■789019 
•789302 
•789585 4 
•789868; 4 
•790151 4 
•790433i 47 
•79071 6i 4 
•790999' 4 
•791281 47 
•791.563 4' 
'•791846! 4 
792128' 4' 
792410! 4' 
7926921 4' 
7929741 47 
•793256' 4- 
•793538' 4 
•793819 469 



10^221226 



469 
469 
469 



•794101 

794383 

794664 

794945 

795227 

795508J 469 

•795789! 468 



469 



10-2209401 
10^220654 \ 
10-220368 
10^220082 
10-219797 
10-2195111 
10-219225' 
10-218940' 
10-218654! 
10-218369 
10-218084 
10-217799 
10-217514 
10-217229 
10-216944 
10-216659! 
10-2163741 
10-216090' 
10-215805' 
10-215521 i 
10-215236 
10-214952 
10-214668 
10-214384 
10-214100 
10-213816 
10-213532 
10-213248'' 
10-212964 
10-212681 I 
10-212397 
10-212114 ■ 
10-211830' 
10-211547 I 
!l0-211264ii 
10-210981 |i 
10-210698 
10-210415 
10-210132 
110-209849 
10-209567 
10-209284 
10-209001 
10-208719 
! 10-208437 
10-208154 I 
10-207872 i 
! 10-207-590 
110-207308 
110-207026 
10-206744 
10-206462 
,10-206181 
!10^205899 
10^205617 
jl0^205336 
|l0-20-5055 
10-204773 
10-204492 
10-204211 



066934 
067010 
067086 
067162 
067238 
067315 
067391 
067467 
067543 
067620 
067696 
067772 
067849 
067925 
068002 
068079 
068155 
068232 
068309 
068386 
068463 
068540 
•068617 
068694 
•068771 
068848 
■068925 



12719- 

12 

1279- 

1279 

I27I9 

12719 

127:9 

1289 

128:9 

128 9 



128 

128 

128 

128 9 

12819 

12819 

1289 

128|9 

1299 

1299 



0690021129 9 
069079il29 9 



129 9 
12919 

1299 
1299 
129|9 
1299 



•069157 

-069234 

•069312 

-069389 

-069467 

•069544 

-069622 12 

-069700!l29l9 

-069777!l30i9 

-0698551130:9 

-069933il30!9 

•07001lil30:9 

-07008913019 

-070167'130:9 

-0702451309 

-070323'130'9 

•070401 130 9 



-070479 
•070558 
•070636 
•070714 
•070793 
•070871 



13019 
130i9 
130 9 



■070950il31 



933066!'60 
932990 59 
932914 58 
932838 57 
932762 56 
932685 55 
932609 54 
932533 53 
932457 52 
932-380 51 
932304 50 
932228 49 
9321-31 48 
932075 47 
931998 46 
93192145 
931845 44 
931768143 
931691 '42 
931614 41 
931-537 '40 
931460 39 
931383 38 
931306 37 
931229 36 
931152 35 
931075 34 
930998 33 
930921 32 
930843 31 
930766 30 
930688:129 
93061l|.j28 
930533127 
930456! 126 
930378:125 
•9-30300' '24 
9-302231123 
■930145122 
■9.30067'21 
■929989;!20 
■92991119 
•929833,18 
•9297-55)117 
•929677j'16 
•92959915 
•929.521:14 
•929442 13 
•929364^12 
•929286' 11 
•929207:10 



Secant. Cotangent. 



Tangent. 



929129' 
9290501 
071028 131J9-928972'! 
928893 I 
•928815:1 
9287361 
928657,1 
928578! 
9284991 
928420 1 



•071107 
•071185 
•071264 
•071343 
•071422 
■071501 
•071580 



Cosecant. 



13119^ 
131|9^ 
13l!9^ 
13l!9- 
13119- 



5S D£6. 



LOGARITHMIC SINES, ETC. 



io 



32 


DEG. 




















' 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Difif. 

100'' 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 







9-724210 




-275790 


9-7957891 


10-204211 


•071580 




9-928420 


60 


1 


9-724412 


337 


•275588 


9-79607O| 468 
9-796351 1 468 


10^203930 


-071658 


132 


9-928342|'59 


2 


9-724614 


337 


•275386 


10^203649 


-071737 


132 


9-928263; 58 


3 


9-724816 


336 


•275184 


9-796632 


468 


10^203368 


-071817 


132 


9-928183157 


4 


9-725017 


336 


•274983 


9-796913 


468 


10-203087 


-071896 


132 


9-928104'56 '■ 


5 


9-725219 


336 


•274781 


9^797194 


468 


10-202806 


-071975 


132 


9-928025 ;55 i 


6 


9-725420 


336 


•274580 


9^797475 


468 


10-202-525 


-072054 


132 


9-927946154 | 


7 


9-725622 


335 


-274378 


9^797755 


468 


10-202245 


-072133 


132 


9-927867|53 


8 


9-725823 


335 


•274177 


9-798036 


467 


10-201964 


•072213 


132 


9-927787 52 


9 


9-726024 


335 


•273976 


9-798316 


467 


10-201684 


•072292 


132 


9-927708 51 


10 


9-726225 


335 


-273775 


9-798596 


467 


10-201404 


•072371 


132 


9-927629 50 


11 


9-726426 


335 


•273574 


9-798877 


467 


10-201123 


-072451 


132 


9-927549 49 


12! 


9-726626 


334 


•273374 


9^799157! 467 


10-200843 


-072530 


132 


9^927470'48 


131 


9-726827 


334 


•273173 


9^799437 


467 


10-200563 1 


-072610 


133 


9^927390 47 1 


14 


9-727027 


334 


•272973 


9^799717 


467 


10-200283 


•072690 


133:9-927310;|46 


15 


9-727228 


334 


•272772 


9^799997 


467 


10-200003 


-072769 


133!9-927231i45 


16 i 


9-727428 


334 


•272572 


9^800277 


466 


10-199723 


-072849 


133|9-927151;|44 


17 1 


9-727628 


333 


-272372 


9^800557 


466 


10-199443 


-072929 


133(9-927071143 


18 


9-727828 


333 


-272172 


9-800836 


466 


10-1991641 


•073009|l33'9^92699r42 1 


19 


9-728027 


333 


•271973 


9-801116 


466 


10-198884 i 


•073089il33'9-926911 41 \ 


20! 


9-728227 


333 


•271773 


9-801396 


466 


10-198604 


-073169133 


9-926831 40 1 


21 1 


9-728427 


333 


-271573 


9-801675 


466 


10-198325 


•073249 


133 9-926751 39 i 


22' 


9-728626 


332 


-271374 


9-801955 


466 


10-1980451 


-073329 


133 9-926671 38 


23 ij 9-728825 


332 


-271175 


9-802234 


466 


10-197766 


-073409 


133 9-926591 37 


24 1 9-729024 


332 


•270976 


9-802513 


465 


10-197487 


•073489 


13319-926511 36 


25 


9-729223 


332 


•270777 


9-802792 


465 


10-197208 


-073569 


134 


9-926431 35 


261 


9-729422 


331 


•270578 


9-803072 


465 


10-196928 


-073649 


134 


9-926351 34 


271 


9-729621 


331 


-270379 


9-803351 


465 


10-1966491 


•073730!l34 


9-926270 33 


28 i 


9-729820 


331 


•270180 


9-803630* 465 


10-1963701 


-073810 


134 


9-926190 32 


291 


9-730018 


331 


-269982 


9-803908 


465 


10-196092 


-073890 


134 


9-926110 31 


30^ 


9-730217i 330 


-269783 


9-804187 


465 


10-195813 


-073971 


134 


9-926029i|30 \ 


31 19-7304151 330 


•209585 


9-804466 


465 


10-195534 


-074051 


134 


9-925949|l29 


32 19-730613! 330 


•269387 


9-804745 


464 


10-195255 


-074132 


134 


9-925868||28 


33 19-730811 


330 


•269189 


9-805023 


464I10-194977 


-074212 


134 


9-925788 27 


34 9-731009 


330 


•268991 


9-805302 


464 110-194698 


-074293 


134 


9-925707 


26 


35; 9-731206 


329 


-268794 


9-805580 


464 |10-194420 


-074374 


134 


9-925626 


25 


36 9-731404 


329 


•268596 


9-805859 


464 10-194141 


-074455 


134 


9-925545 


24 


37^ 9-731602 


329 


•268398 


9-806137 


464 10-193863 


-0745351 135 


9-925465 


23 


38 


9-731799 


329 


•268201 


9-806415 


464|l0-193585 


-074616 


135 


9-925384 


22 


39 


9-731996 


329 


-268004 


9-806693 


464!l0-193307 


•074697 


135 


9-925303 


21 


40 


9-732193 


328 


-267807 


9^806971 


463 10^193029 


•074778 


135 


9-925222 


20 ! 


41 


9-732390 


328 


•267610 


9^807249 


463 10^192751 


-074859 


135 


9-925141 


19 i 


42 J 9-732587 


328 


-267413 


9-807527 


463 |10-192473 


-074940 


135 


9-925060 


18 i 


43 il 9-732784 


328 


•267216 


9-807805 


463I10-192195 


-075021 


135 


9-924979 


1' 1 


44 


9-732980 


328 


•267020 


9-808083 


463 [10-191917 


-075103 


135 


9-924807 


16 


45 


9-733177 


327 


-266823 


9-808361 


463'l0-191639 


•075184 


135 


9-924816 


15 


46 


9-733373 


327 


-266627 


9-808638 


463 10-191362 


-075265 


135 


9-924735 


14 


47 


9-733569 


327 


-266431 


9-808916 


463 10-191084 


-075346 


136 


9-924654 


13 


48 


9-733765 


327 


-266235 


9-809193 


462 110-190807 


-075428 


136 


9-924572 


12 


49 


9-733961 


327 


•266039 


9-809471 


462 10-190529 


-075509 


136 


9-924491 


11 


50 


9-734157 


326 


•265843 


9-8097481 462 10^190252 ! 


-075591 


136 


9^924409 


10 


51 


9-734353 


326 


•265647 


9^810025 


462 10-189975 


•075672 


136 


9-924328 


9 


52 


9-734549 


326 


•265451 


9-810302 


462 10-189698 


•075754 


136 


9-924246 


8 


53 


9-734744 


326 


•265256 


19-810580 


462 10^189420 


•075836 


136 


9-924164 


7 


54 


9-734939 


325 


-265061 


i 9-810857 


462 


10^189143 


•075917 


136 


9-924083 


6 


55 


9-735135 


325 


-264865 


9-811134 


462 


10-188866 


-075999 


136 


9-924001 


5 


56 


9-735330 


325 


•264670 


9-811410 


461 


10-188590 


•076081 


136 


9-923919 


4 


57 


9-735525 


325 


•264475 


9-811687 


461 


10-188313 


-076163 


136 


9-923837 


3 


58 


9-735719 


325 


•264281 


9-811964 


461 


10-188036 


-076245 


1369-923755 


2 


59 


9-7359141 324 


•264086 


19-812241 


461 


10-187759 


•076327 


137 9-923673 


1 


60 


9-736109 


324 


•263891 il 9-812517 


461 !lO-187483 11-076409 


137 9-923591 





Cosine. 


Secant. Cotangent. ' i Tangent. Cosecant.; | Sine. 


' i 



57 D£6. 



74 



LOGARITHMIC SINES, ETC. 



, 


1 

Sine. 


Diff. 

100" 


Cosecant, i Tangent. 


Diff. 

100" 


Cotangent. 


Secant, jj^^-j Cosine. 


1 

'1 





9 


736109 




•263891 9^812517 




10^187483 


•076409! 9-923591 


60 1 


1 


9 


736303 324 


•263697 1 9^812794 


461 


10 


187206 


-076491 137 9-923509 59 


2 


9 


736498 


824 


■268502' 9^813070 


461 


10 


186930 


-076573 137 9-923427 58 


3 


9 


736Q92 


324 


•263308 9^813347| 461 


10 


186653 


|-076655 137 9-923345 57 


4 


9 


736886 


323 


•263114, 9^813628j460 


10 


186877 


'•076737 137;9^923263 56 


5 


9 


737080 


323 


•262920 :9-818899| 460 


10 


186101 


•076819 137 9-923181 55 


6 


9 


737274 


323 


•262726 9-814175 460 


10 


185825 


•076902 137 9-923098 54 


7 


9 


737467 


328 


•262533 9^814452 460 


10 


185548 


-076984:137 9-923016 53 


8 


h 


737661 


323 


•262339 ^•814728 460 


10 


185272 


-077067137 9-922933 52 


9 


19 


737855 


322 


•262145 9-8150041 460 


10 


184996 


•077149 137 9^922851 51 


10 


19 


738048 


322 


•261952 ,9-815279j460 


10 


184721 


•077282 137 9-922768 50 


11 


!9 


738241 


822 


•261759 9-815555 460 


10 


184445 


•077314;138 9^922686 49 


12 1 9 

13 9 


738434 


322 


•261566 9^815831 460 


10 


184169 


•077397 138 9-922603 48 


738627 


822 


•261373 9-8161071 459 


10 


183893 


•077480 188'9^922520 47 


14 9 


738820 


321 


•261180 9-8l6382|459 


10 


183618 


•077562:138 9-922438 46 


15 19 


739013 321 


•260987 9-8166581 459 


10 


183342 


•0776451 138 9^922355 45 


16 


i^ 


739206 321 


•260794 9^816933! 459 


10 


183067 


-077728188 9-922272 44 


17 


9- 


739398 821 


•260602 9-817209; 459 


10 


182791! 


•077811138 9^922189 !43 


18 


9 


739590 321 


•260410 9-817484 459 


10 


182516 


•077894!l38-9-922106:42 


19ij9 


789783 


320 


•260217 9-8177591459 


10 


182241 • 


•077977:138 9-922028 41 


20 


|9 


789975 


320 


•260025 9 -81 8035 1 459 


10 


181965 


•078060'l38 9-921940 40 


21 


I9 


740167 


320 


•259883 9-8183101459 


10 


181690, 


-078143!138!9-921857"39 


22 


19 


740359 


320 


•259641 .9 -81 8585 i 458 


10 


181415 


•078226|l39l9-921774'38 


23 


|9 


740550 


320 


•259450 '9^818860'458 


10 


181140 


-078309 13919-9216911I37 


24 


9 


740742 


819 


•259258 ^•819135;458 


10 


180865 


•078398'139'9-921607'!36 


25 


|9 


740984 


819 


•259066 9^819410: 458 


10 


180590 


-078476 189 9-921524 35 


26 


|9 


741125 


319 


•258875 9-819684 458 


10 


180316 


-078559189 9-921441 34 


27 


9 


741316 


319 


•258684 ,9-819959' 458 


10 


180041 


•078643:139'9^921357|83 
-078726'139!9-921274 32 
•078810'139!9-921190''31 


28 


9 


741508 


319 


•258492 9-820234 458 


10 


179766 


29 


19 


7416991 818 


•258301 9-820508 458 


10 


179492 


30 


l9 


741889' 818 


•258111 9-820783 458 


10 


179217 


•078893 139:9-921107 i30 


1 31 


|9 


742080' 318 


•257920 9-821057; 457 


10 


178943 


-078977139:9-921023' 29 


' 32 


•9 


742271' 818 


•257729 11 9^821882; 457 110 


178668 


■079061 !l89;9-920939 28 


; 33 19 


742462 318 


•257538 i 9-821606 457:10 


178894 


-079144:139!9-920856 27 


34 1! 9 


742652 317 


•257348 '19-821880' 457 10 


178120 


-0T9228!l40i9-920772 26 


35 9 


742842 317 


•257158 19-822154 457 10 


177846 


-079312|l40'9-920688 25 j 


36 


;9 


743083 317 


•256967 i9-822429|457 10 


177571 


-079396140 9-920604 24 


37 


9 


743223 317 


•256777 :i9^822703| 457 110 


177297 


•079480'140!9^920520'28 


38 


9 


743418 317 


•256587 I 9-822977! 457 ;iO 


177023 


•079564 140;9^920436 22 


39 


9 


743602 


316 


•256398 


9-8232501457 110 


176750 


•079648 140 9^920352 21 


40 


9 


743792 


316 


•256208 


9-823524 456 llO 


176476 


•079782 140|9^920268 20 


41 


9 


748982 


316 


•256018' 


9-823798! 456 10 


176202 


•079816 140'9-920184 19 


42 


9 


744171 


316 


•255829 ' 


9-824072! 456 10 


175928 


•079901140 9-920099 18 


43 


9 


*744361 


316 


•255639 I 9-8248451 456 


10 


175655 


•079985il40:9^920015 17 


44 


|9 


744550 


815 


•255450 1 9-824619 456 


10 


175381 


•080069J140 9-919981 16 


45 


9 


744739 


315 


•255261 


9-824898' 456 


10 


175107 


•0801541141 9-919846 15 


46 il 9 


744928 


315 


•255072 , 


9-8251661 456 


10 


174834 


•080238;141 9-919762' 14 


47!i9 


745117 


315 


•254888 


9-825439 


456 


10 


174561 


•080323|l4r9^919677 13 


48 


19 


745306 315 


•254694 


9^825713 


456 


10 


174287 


•080407jl41 9-919593 12 


49 


l9 


745494 314 


•254506 


9^825986 


455 


10 


174014 


•08049214i:9-919508jll 


i 50 


!9 


745683 314 


•254317 '9-826259 


455 


10 


173741 


•080576il41l9^919424;10 


i 51 


I9 


745871 314 


•254129;! 9^826532 


455 


10 


178468 


-080661|14l'9-919339 


9 


1 52 


9 


746060 314 


•253940, j 9^826805 


455 ilO 


173195 


'•080746 141|9-919254 


8 


53 


9 


746248 


314 


•253752 '19-827078 


455 10 


172922 


•080831 14i;9-919169 


7 


54 


'9 


746436 


313 


•253564 '9-827351 


455 ilO 


172649 


•0809l5jl4r9-919085li 6 | 


55||9 


•746624 


313 


•253376 '9-827624 


455 


10 


172876 


•081000 141|9-919000 


5 


56 |9 


746812 813 


•258188!! 9-827897 


455 


10 


172103 


•081085 


142i9^9189l5 


4 


57 19 


•746999 813 


•253001 9-828170 


455 


10 


171880 


81170 


i42i9^9l8830 


3 


i 58 9 


•747187 313 


•252813 !i 9-828442! 454 


10 


171558 


-081255 


142j9-918745 


2 


i 59 j! 9 


•747874 312 


•252626 9-828715' 454:10 


171285 


•081841, 142;9^918659il 1 | 


1 601; 9-747562 812 


•252438 ; 9-828987 454 jlO-171013 


-081426:142!9-918574 

1 1 





j ' i Cosine. 


Secant. Cotangent. • Tangent. ' Cosecant. ! | Sine. 



56 0£G. 



LOGARITHMIC SINES, ETC. 



75 



34 


DE6. 






















1 . ! 

i 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


Diff. 

100" 


Cosine. 


1 





9-747562 




-252438 


9-828987 




10-171013 


-081426 




9-918574 


60" 


1 


9-747749 


312 


•252251 


9-829260 


464 


10 


170740 


•081511 


142 


9-918489 


59 


2 


9-747936 


312 


•252064 


9-829632 


454 


10 


170468 


•081596 


142 


9-918404 


58 


3 


9-748123 


312 


•251877 


9-8298061 464 


10 


170195 


•081682 


142 


9-918318 


57 


4 


9-748310 


311 


•251690 


9-830077 


454 


10 


169923 


•081767 


142 


9-918233 


56 


5 


9-748497' 311 


•251603 


9-830349 


454 


10 


169651 


•0818.53 


142 


9^918147 


55 


6 


9-748683 


311 


•251317 


9-830621 


454 


10 


169379 


•081938 


142 


9-918062 


54 


7 


9-748870 


311 


•251130 


9-830893 


453 


10 


169107 


1-082024 


143 


9-917976 


53 


8 


9-749056 


311 


•250944 


9-831165 


453 


10 


168835 


1-082109 


143 


9-917891 


62 


9 


9-749243 


310 


•260757 


9-831437 


463 


10 


168563 


-082195 


143 


9^917805 


51 


10 


9-749429 


310 


•260571 


9-831709 


453 


10 


168291 


1-082281 


143 


9^917719 


50 


11 


9-749615 


310 


•250385 


9-831981 


463 


10 


168019 


•082366 


143 


9^917634 


49 


12 


9-749801 


310 


•250199 


9-832263 


453 


10 


167747 


•082452 


143 


9-917648 


48 


13 


9-749987 


310 


•250013 


9-832526 


453 


10 


167475 


-082538 


143 


9-917462 


47 


14 


9-750172 


309 


•249828 


9-832796 


453 


10 


167204 


-082624 


14319-917376 


46 


15 


9-750358 


309 


•249642 


9-833068 


453 


10 


166932 


•082710 


143 


9-917290 


45 


16 


9-750543 


309 


•249467 


9-833339 


463 


10 


166661 


•082796 


143 


9-917204 


44 


17 


9-750729 


309 


•249271 


9-833611 


452 


10 


166389 


-082882! 143 


9-917118 i43 


18 


9-750914 


309 


•249086 


9-833882! 452 


10 


166118 


•082968 


144 


9-917032;42 


19 


9-751099 


308 


•248901 


9^834164 


462 


10 


165846 


•083054 


144 


9-916946J41 


20 


9-751284 


308 


•248716 


9-834425 


462 


10 


165576 


•083141 


144 


9-9168-59 


40 


21 


9-751469 


308 


•248631 


9-834696 


452 


10 


166304 


•083227 


144 


9-916773 


39 


22 


9-751654 


308 


•248346 


9-834967 


452 


10 


16.5033 


•083313 


144 


9-916687 


38 


23 


9-751839 


308 


•248161 


9-835238 


452 


10 


164762 


•083400 


144 


9-916600 


37 


24 


9-752023 


308 


•247977 


9-835509 


462 


10 


164491 


•083486 


144 


9-916514 


36 


25 


9-752208 


307 


•247792 


9-835780 


452 


10 


164220 


•083573 


144 


9-916427 


35 


26 


9-752392 


307 


•247608 


9 -830051 


4.52 


10 


163949 


•083659 


144 


9-916341 


84 


27 


9-752576 


307 


.247424 


9^836322 


451 


10 


163678 


•083746 


144 


9-9162-54 


33 


28 


9-752760 


307 


.247240 


9^836693 


451 


10 


163407 


•083833 


144 


9-916167 


32 


29 


9-752944 


307 


.247056 


9-836864 


451 


10 


163136 


•083919 


145 


9-916081 


31 


30 


9-763128 


306 


.246872 


©.837134 


451 


10 


162866 


•084006! 145 


9-916994 


30 


31 


9-753312 


306 


.246688 


9-837405 


451 


10 


162595 


•084093 


145 


9-91.5907 


29 


32 


9-753495 


306 


.246505 


9-837675 


461 


10 


162325 


•084180 


145 


9-916820 


28 


33 


9-753679 


306 


.246321 


9-837946 


461 


10 


1620-54 


•084267 


145 


9-915733 


27 


34 


9-753862 


306 


•246138 


9-838216 


451 


10 


161784 


•084354 


145 


9^915646 


26 


35 


9-754046 


305 


•245964 


9-838487 


461 


10 


161513 


•084441 


145 


9-915559 


25 


36 


9-754229 


305 


•245771 


9-838757 


451 


10 


161243 


•084628 


145 


9^915472 


24 


37 


9-764412 


305 


•245688 


9-839027 


460 


10 


160973 


•084615 


145 


9^916385 


23 


38 


9-754596 


305 


.245406 


9-839297 


460 


10 


160703 


•084703 


145 


9^91-5297 


22 


39 


9-754778 


305 


.246222 


9-839568 


450 


10 


160432 


-084790 


145 


9-915210 


21 


40 


9-754960 


304 


•245040 


9-839838 


450 


10 


160162 


-084877 


146 


9-915123 


20 


41 


9-755143 


304 


•244857 


9-840108 


450 


10 


159892 


-084966 


146 


9-915036 


19 


42 


9-755326 


304 


•244674 


9-840378 


450 


10 


159622 


•085052 


146 


9-914948 


18 


43 


9-755508 


304 


•244492 


9-840647 


460 


10 


159353 


-085140 


146 


9-914860 


17 


44 


9-755690 


304 


•244310 


9-840917 


460 


10 


159083 


•085227 


146 


9-914773 


16 


45 


9-755872 


304 


•244128 


9-841187 


450 


10 


158813 


•086315 


146 


9^914685 


15 


46 


9-756054 


303 


•243946 


9-841457 


450 


10 


158543 


•085402 


146 


9^914598 


14 


47 


9-766236 


303 


•243764 


9-841726 


449 


10 


158274 


-085490 


146 


9-914510 


13 


48 


9-766418 


303 


•243582 


1 9-841996 


449 


10 


158004 


•086578 


146 


9-914422 


12 


49 


9-756600 


303 


.243400 


1 9-842266 


449 


10 


167734 


•085666 


146 


9-914334 


11 


50 


9-756782 


303 


.243218 


9-842535 


449 


10 


157465 


-085754 


146 


9-914246 


10 


51 


9-756963 


302 


.243037 


9-842806 


449 


10 


157195 


•085842 


147 


9-914158 


9 


52 


9-757144 


302 


•242856 ! 9.843074 


449 


10 


156926 


•08-5930 


147 


9-914070 


8 


53 


9-757326 


302 


•242674 


9-843343 


449 


10 


156667 


•086018 


147 


9-913982f 


7 


54 


9-757507 


302 


•242493 


9-843612 


449 


10 


156388 


•086106 


147 


9-913894 


6 


55 


9-757688 


302 


.242312 


[9-843882 


449 


10 


156118 


•086194 


147 


9-913806 


5 


56 


9-757869 


301 


.242131 ji 9-844151 


449 


10 


156849 


•086282 


147 


9^913718 


4 


57 


9-758060 


301 


-241950! 9-844420 


448 


10 


155580 


•086370 


147 


9-913630 


3 


58 


9-758230 


301 


•241770 jl 9-844689 


448 


10 


165311 


-086469 


147 


9-913541 


2 


59 


9-758411 


301 


•2415891 9-844958 


448 


10 


155042 


-086547 


147 


9-913453 


1 


60 


9-758591 


301 


•241409 9-845227 


448 


10 


154773 


•086635 


147 


9^913365! 







Cosine. 


' Secant. 1 Cotangent. 




Tangent. 


Cosecant. 




Sine. 


~^ 



bb SEG. 



76 



35 


DEC. 






















' 


Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diif. 
100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


60 





9-758591 




•241409 


9^845227 




10 


154773 


•086635 




9-913365 


1 


9-758772 


301 


•241228 


9-845496 


448 


10 


154504 


-086724 


147 


9-913276 59 


2 


9-758952 


300 


•241048 


9-845764 


448 


10 


154236 


•086813 


147 


9-913187,58 


3 


9-759132 


300 


•240868 


9-846033 


448 


10 


153967 


-086901 


148 


9-913099:57 


4 


9-759312 


300 


•240688' 9-846302 


448 


10 


153698 


-086990 


148 


9-913010 56 


5 


9-759492 


300 


•240508 h 9-846570 


448 


10 


153430 


•087078 


148 


9-912922155 


6 


9-759672 


300 


•240328 (9 -846839 


448 


10 


153161 


-087167 


148 


9-9128331154 


7 


9-759852 


299 


•240148:19-847107 


448 


10 


152893 


-087256 


148 


9-912744'l53 


8 


9-760031 


299 


•239969 9-847376 


447 


10 


152624 


-087345 


148 


9-912655 52 


9 


9-760211 


299 


-239789 1 9-847644 


447 


10 


152356 


-087434 


148 


9-9l2566:!51 


10 


9-760390 


299 


•239610 !i 9-847913 


447 


10 


152087 


-087523 


148 


9-912477:50 


11 


9-760569 


299 


•239431 ;i9^848181 


447 


10 


151819 


-087612 


148 


9-912388149 


12 


9-760748 


298 


•239252 19-848449 


447 


10 


151551 


-087701 


148 


9-912299:'48 


13 


9-760927 


298 


•239073 i 


9-848717 


447 


10 


151283 


•087790 


149 


9-912210ii47 
9-91212l!46 


14 


9-761106 


298 


•238894 


9-848986 


447 10 


151014 


-087879 


149 


15 


9-761285 


298 


•238715 


9-849254 


447 jlO 


150746 


-087969 


149 


9-912031|i45 


16 


9-761464 


298 


•238536 


9-849522 


447 110 


150478 


-088058 


149 


9-911942 44 


17 


9-761642 


298 


•238358 19-849790 


447 ilO 


150210 


•088147 


149 


9-9.11853I43 


18 


9-761821 


297 


•238179 i: 9-850058 


446 !lO 


149942 


-088237 


149 


9-911763 


42 


19 


9-761999 


297 


•238001 L 9-850325 


446 jlO 


149675 


•088326 


149 


9-911674 


41 


20! 


9-762177 


297 


•2378231:9-8.50593 


446 110 


149407 


-088416 


149 


9-911584 


40 


21 


9-762356 


297 


•237644 9-850861 


446 '10 


149139 


-088505 


149 


9-911495 


39 


22 


9-762534 


297 


•2374661 9-851129 


446 ;10 


148871 


•088595 


149 


9-911405 


38 


23 


9-762712 


296 


•237288 :i 9-851396 


446 10 


148604 


•088685 


149 


9-911315 


37 


24 


9-762889 


296 


•237111! 9-851664 


446 10 


148336 


-088774 


150 


9-911226 


36 


25 


9-763067 


296 


•236933 ii 9-851931 


446 IlO 


148069 


-088864 


150 


9-911136 


'35 


26 


9-763245 


296 


•236755 ii 9-852199 


446 !10 


147801 


-088954 


150 


9-911046 


34 


27' 


9-763422 


296 


•236578 '19-852466 


446 '10 


147534 


•089044 


150 


9-910956'i33 | 


28; 


9-763600 


296 


•236400:, 9-852733 


446 |10 


147267 


-089134 


150|9-910866 32 ! 


29 


9-763777 


295 


•236223 Ii 9-853001 


445 IlO 


146999 


-089224 


150!9-910776:31 \ 


30 


9-763954 


295 


•236046: 19-853268 


445 !10 


146732 


-089314 


150 


9-910686l;30 


31 


9-764131 


295 


-235869:^9-853535 


445 [10 


146465 


-089l04 


150 


9-910596 29 


32 


9-764308 


295 


•235692:; 9-853802 


445 IlO 


146198 


•089494 


15019-910506 28 


33 


9-764485 


295 


•2355151; 9-854069 


445 10 


145931 


-089585 


150|9-910415 27 


34 


9-764662 


294 


•235338;; 9-854336 


445 10 


145664 


•089675 


15019-910325 26 


35 


9-764838 


294 


-235162 ' 9-854603 


445 !10 


145397 


-089765 


151 9-910235 '25 


36 


9-765015 


294 


-234985 ji 9-854870 


445 10 


145130 


-089856 


151|9-910144 24 


37 


9-765191 


294 


•234809! 9-855137 


445 IlO 


144863 


-089946 


151|9-910054 23 I 


38 


9-765367 


294 


•234633 9-855404 


445 IlO 


144596 


-090037 


15l|9-909963 22 | 


39 


9-765544 


294 


•234456:19-855671 


445 IlO 


144329 


-090127 


15l|9-909873 21 • 


40 


9-765720 


293 


•2342801:9-855938 


444 IlO 


144062 


-090218 


151 


9-909782 20 | 


41 


9-765896 


293 


-234104-9-856204 


444 10 


143796 


•090309 


151 


9-909691 19 1 


42 


9-766072 


293 


-233928:! 9-856471 


444 10 


143529 


•090399 


151 


9-909601 18 


43 


9-766247 


293 


•233753 I 9-856737 


444 110 


143263 


•090490 


151 


9-909510 17 


44 


9-766423 


293 


•233577 


9-857004 


444 IlO 


142996 


-090581 


151 


9-909419 16 


45 


9-766598 


293 


-233402 


9-857270 


444 110 


142730 


-090672 


151 


9-909328!l5 


46 


9-766774 


292 


•233226 


9-857537 


444 


10 


142463 


•090763|152 


9-909237{l4 


47 


9-766949 


292 


•233051 


9-857803 


444 


10 


142197 


-090854152 


9-909146113 


48 


9-767124 


292 


•232876 


9-858069 


444 


10 


141931 


-090945 152|9-909055: 12 ! 


49 


9-767300 


292 


•232700 


9-858336 


444 


10 


141664 


-091036!l52l9-908964 11 


50 


9-767475 


292 


•232525 


9-858602 


444 |10 


141398 


•091127 


152:9-908873 10 ; 


51 


9-767649 


291 


•232351 


9-858868 


444 1 10 


141132 


■091219 


152|9-908781 1 9 ! 


52 


9-767824 


291 


•232176 


9-859134i 443 ;10 


140866 


•091310 


15219-908690 8 ■ 


53 


9-767999 


291 


•232001 


9-859400 


443 |10 


140600 


•091401 


152 [9 -908599 


7 


54 


9-768173 


291 


•231827 


9-859666 


443 


10 


140334 


•091493!l52;9-908507 


I 6 


55 


9-768348 


291 


•231652 


9-859932 


443 


10 


140068 


•091584 152I9-908416 


i 5 


. 66 


9-768522 


290 


■231478 


9-860198 


443 


10 


139802 


-091676:153 


9-908324 


4 


57 


9-768697 


290 


•231303 


9-860464 


443 


10 


139536 1 


•091767ll53 


9-908233 


I 3 


58 


9-768871 


290 


•231129 


9-860730 


443 


10 


139270 


-091859153 


9-908141 i 2 1 


59 


9-769045 


290 


•230955 


9-860995 


443 


10 


139005 i 


.-0919511153 


9-908049! 1 


60 


9-769219 


290 


•230781 


9-861261 


443 


10-138739! 


•092042|153 


9^907958l 


Cosine. 




Secant. 


Cotangent. 




Tangent. Cosecant. ! 


Sine. l! ' 1 



54 DE6. 



77 






Sine. 


Diff. 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


^^;i Cosine. 


60 


9-769219 




•230781 


9-861261 




10-138739 


•092042 


!9^907958 


1 


9-769393 


290 


-230607 


9-861527 


443 


10-138473 


-092134 


153 9-907866 59 | 


2 


9-769566 


289 


-280434 


9-861792 


443 


10-138208 


-092226 


153 9-907774! 


58 


3 


9-769740 


289 


•230260 


9-862058 


443 


10-137942 


-092318 


153 9-907682 


57 


4 


9-769913 


289 


-230087 


9-862323 


442 


10-137677 


-092410 


153 9-907590 


56 


5 


9-770087 


289 


-229913 


9-862589 


442 


10-137411 


-092502 


153 9-907498! 


55 


6 


9-770260 


289 


-229740 


9-862854 


442 


10-137146 


-092594 


153 9-907406 


54 


7 


9-770433 


288 


-229567 


9-863119 


442 


10-136881 


-092686 


153 


9-907314 


53 


8 


9-770606 


288 


-229394 


9-863385 


442 


10^136615 


-092778 


154 


9-907222' 


52 


9 


9-770779 


288 


-229221 


9-863650 


442 


10^136350 


-092871 


154 


9-907129, 


51 


10 


9-770952 


288 


•229046 


9-863915 


442 


10-136085 


-092968 


154 


9-907037|!50 


11 


9-771125 


288 


-228875 


9-864180 


442 


10-135820 


•093055 


154 


9-906945ji49 
9-906852148 


12 


9-771298 


288 


-228702 


9-864445 


442 


10-135555 


•093148 


154 


13 


9-771470 


287 


-228530 


9-864710 


442 


10-135290 


-093240 


154 


9-906760;|47 


14 


9-771643 


287 


-228357 


9-864975 


442 


10-135025 


•093333 


154 


9-906667j46 


15 


9-771815 


287 


-228185 


9-865240 


442 


10-134760 


•093425 


154 


9-906575; 


45 


16 


9-771987 


287 


-228013 


9-865505 


441 


10-134495 


•093518 


154 


9-906482 


44 


17 


9-772159 


287 


•227841 


9-865770 


441 


10-134230 


-093611 


154 


9-906389 


43 


18 


9-772331 


287 


•227669 


9-866035 


441 


10-133965 


-098704 


155 


9-906296 


42 


19 


9-772503 


286 


•227497 


9-866300 


441 


10-133700 


•098796 


155 


9-906204 


41 


20 


9-772675 


286 


•227325 


9-866564 


441 


10-183436 


-093889 


155 9-9061111 


40 


21 


9-772847 


286 


•227153 


9-866829 


441 


10-133171 


-098982 


15519-906018 


39 


22 


9-773018 


286 


-226982 


9-867094 


441 


10-132906 


-094075 


155 9-905925 


38 


23 


9-773190 


286 


-226810 


9-867358 


441 


10-132642 


•094168 


155 9-905882 37 


24 


9-773361 286 i 


-226639 


9-867623 


441 


10-132877 


-094261 


l55 9-905789'36 


25 


9-773533 


285 


-226467 


9-867887 


441 


10-132113 


-094355 


155|9-905645Ji35 


i 26 


9-773704 


285 


-226296 


9-868152 


441 


10-131848 


-094448 


155:9-905552134 


1 27 


9-773875 


285 


-226125 


9-868416 


441 


10-131584 


-094541 


155 9-905459 33 


1 28 


9-774046 


285 


-225954 


9-868680| 441 1 


10-131320 


•094634 


155 9-905366 32 


29 


9-774217 


285 


-225783 


9-868945: 440 


10-131055 


•094728 


156 9-905272!|31 


30 


9-774388 


285 


-225612 


9-869209 4401 


10-130791 


-094821 


156 


9-905179:30 


31 


9-774558 


284 


-225442 


9-869473 


440 1 


10-180527 


•094915 


156 


9-905085||29 


32 


9-774729 


284 


•225271 


9-869737 


440 


10-130263 


•095008 


156 


9-904992 |28 


33 


9-774899 


284 


•225101 


9-870001 ! 440 


10-129999 


•095102 


156 


9-904898; 27 


34 


9-775070 


284 


•224930 


9-870265; 440 


10-129735 


-095196 


156 


9-904804 26 
9-904711125 


35 


9-775240 


284 


•224760 


9-870529' 440 


10-129471 


-095289 


156 


36 


9-775410 


284 


•224590 


9-8707931 440 


10-129207 


•095383 


156 


9-9046171 24 


37 


9-775580 


283 


•224420 


9-871057! 440 


10-128943 


•095477 


156 


9-904523' 23 


38 


9-775750 


283 


•224250 


9-871321 440 


10-128679 


•095571 


156 


9-904429:22 


39 


9-775920 


283 


•224080 


9 -8715851 440 


10-128415 


•095665 


157 


9-904885 21 


40 


9-776090 


283 


•223910 


19-871849 


440 


10-128151 


•095759 


157 


9-90424l|'20 


41 


9-776259 


283 


•223741 


19-872112 


440 


10-127888 


•095853 


157 


9-904147i|l9 


42 


9-776429 


283 


•223571 


19-872376 


439 


10-127624 


-095947 


157 


9-904053118 


43 


9.776598 


282 


•223402 


1 9-872640 


439 


10-127360 


-096041 


157 


9-908959117 


44 


9-776768 


282 


•223232 


! 9-872903 


439 


10-127097 


-096136 


157 


9-903864il6 


45 


9-776937 


282 


•223063 


9-873167 


439 


10-126833 


-096230 


157 


9-903770/15 


46 


9-777106 


282 


•222894 


9-873430 


439 


10-126570 


-096324 


157 


9-903676:14 


47 


9-777275 


282 


•222725 


9-873694 


439 


10-126306 


•096419 


157 


9-903581:13 


48 


9-777444 


281 


•222556 


1 9-873957 


439 


10-126043 


•096513 


157 


9-903487l!12 


49 


9-777613 


281 


•222387 


9-874220 


439 


10-125780 


•096608 


157 


9-903392iill 


50 


9-777781 


281 


•222219 


9-874484 


439 


10-125516 


•096702 


158 


9-903298 


10 


1 51 


9-777950 


281 


•222050 


9-874747 


439 


10-125253 


•096797 


15819-903203 


9 


52 


9-778119 


281 


•221881 


9-875010 


439 


10H24990 


•096892|158:9-903108 


8 


53 


9-778287 


281 


•221713 


9-875273 


439 


10-124727 


•096986 158 9^903014 


7 


54 


9-778455 


280 


•221545 


9-875536 


439 


10-124464 


•097081 


158 9-902919 


6 


55 


9-778624 


280 


•221376 


: 9-875800 


439 


10-124200 


•097176 


158i9^902824 


5 


56 


9-778792 


280 


•221208 


9^876063 


438 


10-123937 


•097271 


158'9^902729 


4 


57 


9-778960 


280 


•221040 


9-876326 


438 


10-123674 


-097366 


158'9^902634 


3 


58 


9-779128 


280 


•220872 


' 9-876589 


438 


10-123411 


-097461 


158 9-902539 


2 


59 


9-779295 


280 


•220705 


, 9-876851 


438 


10-123149 


-097556 


159i9-902444 


1 


60 


9-779463 


279 


•220537 


9-877114 


438 


10-122886 


•097651 


159,9-902349 





'ZJl 


Cosine. 


! Secant. Cotangent. 


1 


Tangent. 


1 Cosecant. 


1 Sine. 



53 DE6. 



78 



37 


DEG. 




















1 . 

1 


Sine. 


Diff. 

100" 


Cosecant. 


Tangent. 


fJo^; Cotangent. 


Secant. 


Diff. 
100" 


cosine. Ij ' 





9-779463 




•220537 


9-877114 


,10-122886 


-097651 




9-902349 60 


1 


9-779631 


279 


•220369 


9-877377 


438 ilO-122623 


•097747 


159 


9-902253 .59 J 


2 


9-779798 


279 


•220202 


9-8776401 438 ilO-122360 


-097842 


159 


9-9021.58-58' 


3 


9-779966 


279 


•220034 


9-877903; 438 |10-122097 


-097937 


159 


9-902063 57 


4 


9-780133 


279 


•219867 


9-878165: 438 10-121835 


-098033 


159 


9-901967 56 


5 


9-780300 


279 


•219700 


9-8784281438 10-121572 


-098128 


159 


9-901872 55 


6 


9-780467 


278 


•219533 


I9-878691, 438 10-121309 


-098224 


159 


9-901776 -54 


7 


9-780634 


278 


•219366 


19-878953^438 110-121047 


-098319 


15919-901681 .53 


8 


9-780801 


278 


•219199 


j 9-879216; 437 10-120784 


-098415 


159i9-901585 52 


9 


9-780968 


278 


•219032 


9-879478' 437 :10-120522 


-098510 


159 


9-901490 51 1 


10 


9-781134 278 


■218866 


19-879741! 437 ,10-120259 


-098606 


1.59 


9-901394 50 


11 


9-781301 278 


•218699 11 9-880003 437 ilO-119997 


•098702 


160 


9-901298 49 


12 


9-781468 277 


-218-532 ll 9-880265 437 1 10-119735 


-098798 


160 


9-901202 148 


13 


9-781634 277 


•218366 i| 9-880.528 437 10-119472 


-098894 


160J9-901106 !47 1 


14 


9-781800 277 


-218200119-8807901 437 ilO-119210 


-098990 


16019-901010 146 ! 


15 


9-781966 277 


-218034 


i 9-8810-52 437 10-118948 


-099086 


16019-900914 :45 1 


16 


9-782132 277 


-217868 


i 9-881314! 437 !lO-118686 


-099182 


160'9-900818 :44 


17 


9-782298; 277 


-217702 


J9-881576i 437 10-118424 


•099278 


160 9-900722 43 


18 


9-782464 


276 


•2175-36 


1 9-881839 437 10-118161 


-099374 


160 9-900626 42 


19 


9-782630 


276 


•217370 


|9-88210li437il0-117899 


•099471 


I60I9-9OO.529 41 


20 


9-782796 


276 


•217204 


9-882363'437|10^117637 


•099567 


160'9-900433 40 


21 


9-782961 


276 


-217039 


9-882625 437 |10-117375 


-099663 


161:9-900337 39 


22 


9-783127 


276 


-216873 


9-8828871436 110-117113 


-099760 


161i9-900240 38 


23 


9-783292 


276 


•216708 


9-883148 436 110-116852 


-099856 


161I9-900144 37 


24 


9-783458 


275 


-216542 


9-883410 


436 10-116590 


-099953 


161 19-900047 36 


25 


9-783623 


275 


•216377 


9-883672 


436|l0-116328 


-100049 


16l!9-899951 35 i 


26 


9-783788 


275 


•216212 


9-883934 


436 110-116066 


-100146 


16110-8998.54 34 i 


27 


9-783953 


275 


•216047 


9-884196 436 110-115804 


-100243 


161 


9-899757 33 ! 


28 


9-784118 


275 


•215882 


9-884457 


436 110-115543 


-100340 


161 


9-899660 32 


29 


9-784282 


275 


•215718 


9-884719 


436 110-115281 


■100436 


161 


9-899-564 31 ' 


30 


9-784447 


274 


•215553 


9-884980 


436il0-115020 


-100533 


161 


9 -899467 '^SO | 


31 


9-784612 


274 


•215388 19-885242 


436 110-1147-58 


100630 


162 


9-899370:29 1 


32 


9-784776 


274 


•215224} 9-885-503 


436 |10-114497 


-100727 


162 


9-899273ii28 i 


33 


9-784941 


274 


•215059 i! 9-885765 


436 !l0-ll4235 


•100824 


162 


9-899176:'27 1 


34 


9-785105 


274 


•214895 


1 9-886026 


436 jlO-113974 


-100922 


162 


9-899078.26 1 


35 


9-785269 


274 


•214731 


1 9-886288 


436 |10-113712 


-101019 


162 


9-898981!!25 ! 


36 


9-785433 


273 


•214567 


9-886549 


436 ilO-113451 


•101116 


162 


9-898884 


24 


37 


9-785597 


273 


•214403 


9-886810 436 ilO-113190 


•101213 


162 


9-898787 


23 1 


38 


9-785761 


273 


•214239 


9-887072 


435 10-112928 


101311 


162 


9-898689 


22 j 


39 


9-785925 


273 


•214075 


9-887333 


435 110-112667 


•101408 


162 


9-898592 


21 1 


40 


9-786089 


273 


•213911 


9-887594 


435 10-112406 


•101506 


16219-898494 


20 1 


41 


9-786252 


273 


•213748 


9-887855 


435 10-112145 


•101603 


163 


9-898397 


19! 


42 


9-786416 


272 


•213584 


9-888116 


435 10-111884 


•101701 


163 


9-898299 


18 


43 


9-786579 


272 


•213421 


9-888377 


435 110-111623 


•101798 


163 


9-898202 


17 


44 


9-786742 


272 


•213258 


9-888639 


435 10-111361 


•101896 


163 


9-898104 


16 


45 


9-786906 


272 


•213094 


9-888900 


435 10-111100 


•101994 


163 


9-898006 


15 


46 


9-787069 


272 


•212931 


9-889160 


435 10-110840 


•102092 


163 


9-897908 


14 


47 


9-787232 


272 


-212768 


9-889421 


435 10-110579 


•102190 


163 


9-897810 


13 


48 


9-787395 


271 


•212605 


9-889682 


435 110-110318 


•102288 


163 


9-897712 


12 


49 


9-787557 


271 


•212443 


9-889943 


435 10-110057 


•102386 


163 


9-897614 


11 


60 


9-787720 


271 


•212280 


9-890204 


435 110-109796 


•102484 


163 


9-897516 


10 


51 


9-787883 


271 


•2121171 


9-890465 


435 jlO-109535 


•102582 


163 


9-897418 


9 


52 


9-788045 


271 


•2119-55 ! 


9-890725 


435 10-109275 


-102680 


164 


9-897320 


8 


53 


9-788208 


271 


•211792 


9-890986 


434 10-109014 


-102778 


164 


9-897222 


7 


54 


9-788370 


271 


•211630 


9-891247 


434 


10-108753 


-102877 


164 


9-897123 


6 


55 


9-788532 


270 


•211468 


9-891507 


434 


10-108493 


•102975 


164 


9-897025 


5 


56 


9-788694 


270 


•211306 


9-891768 


434 


10-108232 


•103074 


164 


9-896926 


4 


57 


9-788856 


270 


•211144 


9-892028 


434 


10^107972 


•103172 


164 


9-896828 


3 


58 


9-789018 


270 


•210982 


9-892289 


434 


10-107711 


103271 


164l9-896729i 


2 


59 


9-789180 


270 


-210820 


9-892549 


434 


10-107451 


•103369 


164 


9-896631 


1 


60 


9-789342 


270 


-210658 


9-892810 


434 


10-107190 


•103468 


164 


9-896532 





Cosine. 


Secant. ' Cotangent. 


Tangent. 


Cosecant. 




Sine. 


' i 



52 D£G. 



LOGARITHMIC SINES, ETC. 



79 



38 


DEG. 




















' 


Sine. 


DifF. 
100" 


Cosecant. 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


' 





9-789342 




•210658 


9^892810 




10-107190 


-103468 




9-896632 


60" 


1 


9-789604 


269 


-210496 


9-893070 


434 


10-106930 


•103567 


164 


9-8964331 


59 


2 


9-789665 


269 


•210336 


9-893331 434 


10-106669 


•103665 


165 


9-896335, 


58 


3 


9-789827 


269 


-210173 


9-893691 


434 


10-106409 


•103764 


165 


9-8962861 


57 


4 


9-789988 


269 


•210012 


9^893851 


434 


10-106149 


•103863 


166 


9-896137 56 


5 


9-790149 


269 


•209861 


9-894111 


434 


10-106889 


•103962 


165 


9-896038 65 


6 


9-790310 


269 


•209690 


9-894371 


434 


10-106629 


•104061 


165 


9-895939;54 


7 


9-790471 


268 


•209629 


9-894632 


434 


10-106368 


-104160 


165 


9-896840: 


53 


8 


9-790632 


268 


-209368 


9-894892 


434 


10-105108 


-104269 


166 


9-8957411 


52 


9 


9-790793 


268 


-209207 


9-896162 


433 


10-1048481 


-104369 


166 


9-895641; 


51 


i 10 


9-790964 


268 


•209046 


9-895412 


433 


10-104688 


-104458 


165 


9-895642^ 


50 


J 11 


9-791116 


268 


•208885 


9-895672 


433 


10-1043281 


-104567 


166 


9-8964431 


49 


12 


9-791276 


268 


•208725 


9-895932 


433 


10-104068 i 


-104667 


166 


9-895343 !48 | 


13 


9-791436 


267 


•208564 


9-896192 


433 


10-103808 


-104766 


166 


9-895244 


47 


14 


9-791696 


267 


•208404 


9-896462 


433 


10-103548 


-104855 


166 


9-895145 


46 


15 


9-791767 


267 


•208243 


9-896712 


433 


10-103288 


-104966 


166 


9-895046 


45 


16 


9-791917 


267 


•208083 


9-896971 


433 


10-103029 


•10^50.56jl66 


9-894946 


44 


17 


9-792077 


267 


•207923 


9-897231 


433 


10-102769 


•1061541166 


9-894846 


43 


18 


9-792237 


267 


•207763 


9-897491 


433 


10-102609 


•106254!l66 


9-894746 


42 


19 


9-792397 


266 


•207603 


9-897761 


433 


10-102249 


•1063541166 


9-894646 


41 


1 20 


9-792567 


266 


•207443 


9-898010 


433 


10-1019901 


-106464ll66 


9-894646 


40 


21 


9-792716 


266 


•207284 


9-898270 


433 


10-101730 1 


-106554 166 


9-894446 


39 


22 


9-792876 


266 


•207124 


9-898630 


433 


10-101470 


•106664 167 


9-894346 


38 


i 23 


9-793036 


266 


•206965 


9-898789 


433 


10-101211 


•105754!167 


9-894246 


37 


, 24 


9-793196 


266 


•206805 


9-899049 


433 


10-100961 


•10-5854!l67J9-894146 


36 


i 25 


9-793364 


265 


•206646 


9-899308 


432 


10-100692 


•105954 


167 


9-894046 


35 


! 26 


9-793614 


265 


•206486 


9-899668 


432 


10-100432 


-1060.54 


167 


9-893946 


34 


1 27 


9-798673 


266 


•206327 


9-899827 


432 


10-100173 


-106154 


167 


9-893846 


33 


i 28 


9-793832 


265 


•206168 


9^900086 


432 


10-099914 


-106256 


167 


9-893746 


32 


1 29 


9-793991 


266 


•206009 


9-900346 


432 


10-099664 


-106355 


167 


9-893645 


31 


1 30 


9-794160 


266 


•205850 


9^900606 


432 


10-099395 


•106466 


167 


9-893544 


30 


31 


9-794308 


264 


•205692 


9-900864 


432 


10-099136 


•106556 


167 


9-893444 


29 


32 


9-794467 


264 


•206633 


9-901124 


432 


10-098876 


•106667 


168 


9-898343 


28 


33 


9-794626 


264 


•206374 


9-901383 


432 


10-098617 


-106767 


168 


9-893243 


27 


34 


9-794784 


264 


•206216 


9-901642 


432 


10-098368 


-106858 


168 


9-893142 


26 


35 


9-794942 


264 


•206058 


9-901901 


432 


10-098099 


■106959 


168 


9-893041 


25 


36 


9-795101 


264 


•204899 


9-902160 


432 


10-097840 


•107060 


168 


9-892940 


24 


37 


9-796259 


264 


•204741 


9-902419 


432 


10-097581 


•107161 


168 


9-892839 


23 


38 


9-795417 


263 


•204583 


9-902679 


432 


10-097321 


•107261 


168 


9-892739 


22 


39 


9-796675 


263 


•204426 


9-902938 


432 


10-097062 


-107362 


168 


9-892638 


21 


40 


9-796733 


263 


•204267 


9-903197 


432 


10-096803 


•107464 


168 


9-892536 


20 


41 


9-795891 


263 


•204109 


9-903466 


432 


10-096645 


•107566 


168 


9-892435 


19 


42 


9-796049 


263 


•203951 


9-903714 


431 


10-096286 


•107666 


169 


9-892334 


18 


i 43 


9-796206 


263 


•203794 


9-903973 


431 


10-096027 


•107767 


169 


9-892233 


17 


44 


9-796364 


263 


•203636 


9-904232 


431 


10-096768 


-107868 


169 


9-892132 


16 


45 


9-796521 


262 


•203479 


9-904491 


431 


10-096609 


•107970 


169 


9-892030 


15 


46 


9-796679 


262 


•203321 


9-904760 


431 


10-096250 


-108071 


169 


9-891929 


14 


47 


9-796836 


262 


•203164 


9-905008 


431 


10-094992 


-108173 


169 


9-891827 


13 


48 


9-796993 


262 


•203007 


9-905267 


431 


10-094733 


•108274 


169 


9-891726 


12 


49 


9-797160 


262 


•202850 


9^906526 


431 


10-094474 


•108376 


169 


9-891624 


11 


50 


9-797307 


262 


•202693 


9-906784 


431 


10-094216 


•108477 


169 


9-891523 


10 


51 


9-797464 


261 


•202536 


9-906043 


431 


10-093957 


•108579 


169 


9-891421 


9 


52 


9-797621 


261 


•202379 


9-906302 


431 


10-093698 


-108681 


170 


9-891319 


8 


53 


9-797777 


261 


•202223 


9-906660 


431 


10-093440 


-108783 


170 


9-891217 


7 


54 


9-797934 


261 


•202066 


9-906819 


431 


10-093181 


-108885 


170 


9-891115 


6 


65 


9-798091 


261 


•201909 


9-907077 


431 


10-092923 


•108987 


170 


9-891013 


5 


56 


9-798247 


261 


•201753 


9-907336 


431 


10-092664 


-109089 


170 


9-890911 


4 


57 


9-798403 


261 


•201597 


9-907694 


431 


10-092406 


-109191 


170 


9-890809 


3 


58 


9-798560 


260 


•201440 


9-907862 


431 


10-092148 


-109293 


170 


9-890707 


2 


59 


9-798716 


260 


•201284 


9-908111 


431 


10-091889 


•109395 


170 


9-890605 


1 


60 


9-798872 


260 


•201128 


i 9-908369 


431 


10-091631 


•109497 


170 


9-890603 





' 


Cosine. 




Secant. 


1 Cotangent. 




Tangent. 


Cosecant. 




Sine. 


' 



51 DEO. 



80 



LOaARITHMIC SINES, ETC. 



39 


DEC. 




















■ 


Sine. 


Diff. , 

100" 


Cosecant. 1 Tangent. 1 ^; 


Cotangent. 


Secant. 


iT, ^^^-• 


160 





9-798872 


i 


•201128!: 9-9083691 


10-091631 


•109497 


9-890503 


1 


19-799028 


260 


•200972 


9-908628! 430 


10-091372 


-109600 


170 9-890400! 59 


2 


19-799184 


260! 


•200816 


9-9088861 430 


10-091114 


-109702 


171 9-890298, .58 


3 


'9-799339 


2601 


•200661 


9-9091441 430 


10-090856 


•109805 


17i:9-890195ii57 


4 


! 9-799495 


259 


•200505 i 


9-909402 480 


10-090598 


-109907 


171 9-890093i;56 


5 


i 9-799651 


259; 


•200349 1 


9-909660 


430 


10-090340 


-110010 


171 9-889990; 55 


6 


9-799806 


259! 


•200194!: 9-909918 


430 


10-090082 


•110112 


17119-889888:54 


7 


19-799962 


259! 


•200038 ! 9-910177 


430 


10-089823 


•110215 


171i9-889785:53 ! 


8 


9-800117 


259 


•199883 : 


9-910435 


430 


10-089565 


•110818!l71 9-889682 -52 


9 i: 9-800272 


259 


•199728 


9-9106931 430 


10-089307 


-110421 11719-889579 51 


10 li 9-800427 


258 


•199573 


9-9109511430 


10-089049 


-1105231171 9-889477 50 


11 li 9-800582 


258 


•199418 


9-911209 430 


10-088791 


■1106261171 9-889374 49 


12 


i 9-800737 


258 


•199263 


9-911467 430 


10-088533 


■110729;172 9-889271 ,48 


13 


9-800892 


258 


•199108 


9-911724 430 


10-088276 


-110832il72'9-889168i47 


14 


9-801047 


258 


•198953 


9-911982! 480 


10-088018 


-110936 172.9-889064 46 


■ 15 


I 9-801201 


258 


•198799 


9-912240' 430 


10-087760 


-1110391172 9-888961)45 


16 ii 9-801356 


258 


•198644 : 


9-912498 430 


10-087502 


-111142 172. 9-888858; 44 


17! 9-801511 


257 


•198489 


9-912756 430 


10-087244 


-111245!l72 9-888755 43 


18 : 9-801665 


257 


•198335 


9-913014 


430 


10-086986 


-111349 172 9-888651 42 


19 9-801819 


257 


•198181 


9^913271 


430 


10-086729 


-111452 172 19-888548: 41 


20 ii 9-801973 


257 


•198027 


9-913529 


429 


10-086471 


-111556!l72j9-888444|40 


21; 9-802128 


257 


•197872 


9-913787 


429 


10-086213 


-111659;i73!9-888341ii39 


22 : 9-802282 


257 


•197718 


9-914044 


429 


10-085956 


-111763173:9-888237 ^38 


231 19-802436 


256 


•197564 


9-914302 


429 


10-085698 


•111866il73;9-888134 37 


24:19-802589 


256 


•197411 


9-914560 


429 


10-085440 


-111970il73:9-888030i36 


25' 9-802743 


256 


•197257 


9-9148171429 


10-085183 


•112074173 9-887926' 35 


26 9-8028971256 


•197103 


9-915075: 429 


10-084925 


-112178 173 9-887822 134 


27 i 9-8080501 256 


•196950 


i9-915332i429 


10-084668 


•112282 173!9-887718;|33 


28 9-803204! 256 


•196796 


9-915590! 429 


10-084410 


•112386173;9-887614i'32 


29 i 9-803357 256 


•196643 


9-915847 429 


10084153 


-112490 173 9-887510' 31 


30 9-803511 255 


•196489 1:9-916104: 429 


10-083896 -112594'l73!9-887406''30 I 


31 i, 9-803664 255 


•196336 


I 9-916362! 429 


10-083688 -112698:l74:9-887302i29 | 


32 1 9-803817 255 


•196183 


9-916619! 429 'lO-OSSSSl :-112802;l74|9-887198 28 i 


33; 9-803970 255 


-196030 


9-916877! 429 10-088123 


•112907:174 


9-887093 27 ! 


34 1:9-804123 255 


•195877 


9-917134i 429 ;10-082866 


-113011 


174 


9-8869891126 ! 


35 i 9-804276 255 


-195724 


9-917391 429 ilO-082609 


•113115 


174 


9-886885:25 * 


36 ii 9-804428! 254 


•195572 


9-917648 429 10-082352 


•113220 


174 


9-886780 |24 ! 


37 ,9-8045811 254 


•195419 


9-917905 429 10-082095 


-1133-24!l74 


9-886676123 j 


38 9-8047341254 


•195266 


9-918163! 429 10-081837 


•118429!l74:9-886571 22 1 


39 „ 9-804886^ 254 


-195114 


9-918420: 429 10-081580 


•118584174 9-886466 21 


40 || 9-805039: 254 


-194961 


9-918677 429 10-081823 


'-118638174i9-886362|20 


41 119-805191! 254 


-194809 


9-918934 429 10-081066 


,-113743 175i9-886257.19 1 


42 9-80534?' 254 


-194657 


19-919191 428 10-080809 


113848;175!9-886152i|18 ! 


43 9-805495 253 


-194505 


9-919448 428 ilO-080552 


-118958!175i9-886047!:17 ! 


44 9-805647! 253 


-194353 


9-919705 428 '10-080295 


•114058175 9-885942116 1 


45 ; 9-805799 253 


-194201 


9-919962 428 10-080038 


-114163175:9-885887 !15 ! 


46 19-805951 253 


•194049 


9-920219 


428 10-079781 


-114268 1759-885732 


14 ! 


47 1 9-806103 253 


•193897 


9-920476 


428 10-079524 


-114373 175 9-885627 


13 ' 


48 i| 9-806254 253 


-193746 


9-920733 


428:10-079267 


-114478175 9-8855221 


12 ' 


49:19-806406 253 


-193594 


9-920990 


428 !lO-0790lO 


■114584 175 9-885416; 


11 i 


50 9-806557.252 


-193443 


19-921247 


428 :iO-078753 


-114689175 9-885311 


10 i 


51 ,9-806709 252 


•193291 


! 9-921503; 428 ilO-078497 


-114795176 9-885205 


9 ' 


52 9-806860 252 


•193140 


9-921760 428110-078240 


-114900176 9-885100 


8 


53 ii 9-807011: 252 


•192989 


9-922017 428 !lO-077983 


-115006 1769-884994 


7 ; 


54:19-807163 252 


•192837 


9-922274 428 i] 0-077726 


-115111 


176 9-884889 


6 ! 


55: 9-807314 252 


•192686 


9-922530! 428 ilO-077470 i -115217 


176 9-884783 


5 1 


56 9-807465 252 


•192535 


9-922787; 428 10-077213 -115323 


176 9-884677 


4 i 


57: 9-807615 251 


•192385 i 9-923044 428 10-076956 i'-115428 


176 9-884572! 


3 


58 9-807766 251 


-192234 9-923300 428 10-076700 


-115534il76 9-884466J 


2 


59: 9-807917 251 


•192083 ii 9^923557; 428 10-076443 


-11564011769-884360; 


1 ! 


60 9-808067 251 


•191933 I' 9^923813 428 10-076187 


-115746jl76;9-884254j 


! 


' Cosine. ; 


Secant. Cotangent. 1 Tangent. 


Cosecant. 1 Sine. 1 


~l 



50 DEG. 



LOGARITHMIC SINES, ETC. 



81 



40 


DEG. 




















' 


Sine. 


Diff. 1 
100" 


Cosecant. 


Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


^, Cosine. 


' 





9-808067 1 


•191933 


9-923813 




10-076187 


•115746 


9-884254 


60 


li 


9-808218 251 


•191782 


9-924070 


428 


10-075930 


•115852 


177 9-884148 


59 


2i 


9-808868 251 


•191632 


9-924327 


428 


10-075673 


•115958 


177 9-884U42 58 


3j 


9-808519,251 


•191481 


9-924583 


428 


10-075417 


•116064 


177i9-883936(,57 
177,9-883829 56 


4| 


9-808669, 250 


•191331 


9-924840 


427 


10-075160 


•116171 


5! 
6! 


9-808819 250 


•191181 


9-925096 


427 


10-074904 


•116277 


177 9-883723 55 


9-808969 250 


•191031 


9-925352 


427 


10-074648 


•116383 


177 9-8836171,54 


71 


9-809119 250 


•190881 


9^925609 


427 


10-074391 


•116490 


177 9-883510 53 


81 


9-809269 


250 


•190731 


9-925865 


427 


10-074135 


•116596 


177 9-883404,52 


^ 


9-809419 


250 


•190581 


9-926122 


427 


10-073878 


•116703 


177 9-883297 |51 


10 


9-809569 


249 


•190431 


9-926378 


427 


10^073622 


•116809 


178,9-8831911 50 


11 


9-8097181 249 


•190282 


9-926634 


427 


10-073366 


•116916 


178 9-883084149 


12 


9-809868 249 


•190132 


9-926890 


427 


10-073110 


-117023 


17819-882977148 


13 


9-810017 


249 


•189983 


9-927147 


427 


10-072853 


-117129 


178,9-882871 147 


14 1 


9-810167 


249 


•189833 


9-9274031 427 


10-072597 


-117236 


178.9-882764I146 


15! 


9-810316 


249 


•189684 


9-9276591 427 


10-072341 


-117343 


178'9-882657!'45 


lb| 


9-810465 


248 


•189535 


9-927915! 427 


10-072085 


-117450 


178 9-882550'!44 


1' 


9-810614 


248 


•189386 


9-9281711427 


10-071829 


-117557 


178 9-882443 '4.3 


18 1 


9-810763 


248 


•189237 


9-9284271 427 


10-071573 


•117664 


178'9-882336 42 


19! 


9-810912 


248 


•189088 


9-9286831 427 


10-071317 


-117771 


179;9-882229;'4l 


201 


9-811061 


248 


•188939 


9-928940| 427 


10-071060 


•117879 


179!9-88212l!l40 


211 
22 


9-811210 


248 


•188790 


9-929196! 427 


10-070804 


-117986 


179i9-882014!39 


9-811358' 248 


•188642 


9 •929452! 427 


10-070548 


•118093 


179 9-881907:38 


23 


9-811507: 247 


•188493 


9-929708! 427 10-070292 | 


-118201 


179!9-881799:i37 


24 9-811(J55 247 


•188345 


9-929964! 427 


10-070036 


•118308 


179|9-881692136 


25 1 9-811804 247 


•188196 


9-930220| 427 


10-069780 


-118416 


179!9-881584!35 


26 1 9 -81 1952; 247 


•188048 


9-930475427 


10-069525 


•118523 


179 9-881477 34 


27ily-812100'247 


•187900 


9-930731! 427 


10-069269 


•118631 


179 


9-881369il33 


28 !i 9-812248: 247 


•187752 


9-930987 426 


10-069013 


•118739 


179 


9-8812611132 


29 |l 9-8123961 247 


•187604 


9-9312431 426 


10-068757 


•118847 


180 


9-881153:31 


30 !! 9-8125441 24(3 


•187456 


9-931499! 426 


10-068501 


•118954 


180 


9-881046'30 


31, 


9-812692' 246 


•187308 


9-931755 426 


10-068245 


•119062 


180 


9-880938|l29 


32 


9-812840! 246 


•187160 


9-932010 426 


10-067990 


•119170 


180 


9 -8808301 128 


33 


9-812988 


246 


•187012 


! 9-9322661 426 


10-067734 


•119178 


180 


9 -880722' l27 


34 


9-813135 


246 


•186865 


i 9-9325221 426 


10-067478 


■119387 


180!9-880613i!26 | 


35 


9-813283 


246 


•186717 


9-932778; 426 


10-067222 


•119495 


180 


9-880505 


25 


36 


9-813430 


246 


•186570 


1 9-933033 426 


10-066967 


•119603 


180 


9-880397 


24 


37 


9-813578 


245 


•186422 


i 9-933289' 426 


10-066711 


•119711 


180 


9-8802891 


23 


38 


9-813725 


245 


•186275 


1 9^933545: 426 


10-066455 


•119820 


181 


9-880180! 


22 


39 


9-813872 


245 


•186128 


! 9-933800! 426 


10-066200 


•119928 


181 


9-8800721 


21 


40 


9-814019 


245 


•185981 


I 9-934056i 426 


10-065944 


-120037 


181 


9-879963: 


20 


41 


9-814166 


245 


•185834 


9-934311! 426 


10-065689 


•120145 


181 


9-879855^ 


19 


42 


9-814313 


245 


•185687 


9-934567 426 


10065433 


•120254 


181 9-8797461 


18 


43 


9-814460 


245 


•185540 


9-934823 426 


10-065177 


•120363 


181 9-879637 Il7 1 


44 


9-814007 


244 


•185893 


9-935078 426 


10-064922 


•12047lil8l|9-879529ill6 | 


45 


9-814753 


244 


•185247 


9-935333 


426 


10-064667 


•120580 


1819-879420 


15 


46 


9-814900 


244 


•185100 


9-935589 


426 


10-064411 


•120689 


1819-879311 


14 


47 


9-815046 


244 


•184954 


9-935844 


1 426 


10-064156 


•120798 


18l|9-879202 


13 


48 


9-815198 


244 


•184807 


9-9361001 426 


10-063900 


•120907 


18219-879093 


12 


49 


9-815339 


244 


•184661 


9-9363551 426 


10-063645 


•121016 


182!9-878984!|ll 


50 


y-81548o 


244 


•184515 


9-936610 426 


10063390 


•121125 


18219-878875 llO 


51 


9-815632 


243 


•184368 


9-9368661426 


10-063134 


•121234 


182|9-878766' 


9 


52 


9-815778 


243 


•184222 


9-937121 


426 


10-062879 


•121344 


182,9-878656 


8 


63 


9-815924 


243 


•184076 9-937376 


426 


10-062624 


•121453 


182'9-878547 


7 


54 


9-816069 


243 


•183931 9-937632 


425 


10-062368 


•121562 


18219-878438 


6 


00 


19 816215 


243 


•183785 9-937887 


425 


10-062113 


•121672 


1829-878328 


5 


56j!9-8]6::]61 


243 


•183689 9-938142 


425 


10-061858 


•121781 


182[9-878219 


4 


57 ;i 9-816507 


243 


•183493 9-938398 


!425 


10061602 


•121891 


183:9-878]09' 


3 


58 |i 9-816652 


U42 


•183348 9-938653' 425 


10061347 


j-122001 1183 9-877999 


2 


59 'l 9-816798 


1242 


•183202 9-938908 425 


|10-061092 


-122110 183 9-877890 


1 


1 60 


I 9 -8: 6948! 242 


•183057 9-9391631 425 ilO-060837 


•122220 188|9^877780 





, ' 


Cosine. ' 


>oi:r.it. Cotangent. ! ' Taugent. 


Cosecant. I [ Sine. 



49 



82 



LOGARITHMIC SINES, ETC. 



41 


DES. 























Sine. 


Diff. 1 

100" 1 


1 
Cosecant. ! 

■ 


Tangent. 


Diff. 
IOC' 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


1 , 


9-816943 




•183057 1 9-9391631 


10^060837 


•122220 




9-877780 60 


1 


9-817088 


242 


•182912 :! 9-939418 425 10-060582 


•122330 


183 


9-877670 59 


2 


9-817233 


242 


•182767 ; 9-939673; 425 10-060327 


•122440 


183 


9-877560 58 


3 


9-817379 


242 


•182621: 19-939928; 425 


10-060072 


•122550 


183 


9-877450 57 


4 


9-817524 


242 


•182476:1 9-94-01831 425 


10-059817 


•122660 


183 


9-877340' -56 


5 


9-817668 


241 


•182332 19-9404381 425 


10-059562 


•122770 


183 


9-877230 55 


6 


9-817813 


241 


•182187 19-9406941425 


10-059306 


-122880 


184;9-877120 54 i 


7 


9-817958 


241 


-182042 9-9409491 425 


10-059051 


-122990 


184 


9-877010 53 


8 


9-818103 


241 


•181897 9-941204{425 


10-058796 


-12310], 


184 


9-876899 52 


9 


9-818247 


241 


•181753 9-941458' 425 


10-058542 


-123211 


1«4 


9-876789 51 


10 i 


9-818392 


241 


•181608 i 9-9417141 425 10-058286 I 


-123322 


184 


9-876678':50 


11 1 


9-818536 


241 


•181464 ' 9-9419681 425 |10-058032 


-123432 


18419-876568' 49 i 


12 1 


9-818681 


240 


-181319 '\ 9-942223! 425 jlO-057777 | 


-123543|184i9-876457;48 \ 


13 i 


9-818825 


240 


•181175.: 9-942478 425 


10-057-522 


-123653 


184,9-876347 47 


14 


9-818969 


240 


•181031 j 9-942733I 425 


10-057267 


■123764 


184:9-876236''46 


15 


9-819113 


240 


•18088711 9-942988 425 


10-057012 


-123875 


185:9-876125 45 


16 


9-819257 


240 


•180743^ 9-9432431425 


10-056757 


■123986 


185!9-876014 44 


17! 


9-819401 


240 1 


•180599 9-9434981 425 


10-056-502 


•124096 


185 j9 -875904; 43 


18 1 


9-819545 


2401 


•180455 9-943752:425 


10-056248 


-124207 


l85 9-875793i42 


19 


9-819689 


239 


•180311 ' 9-944007! 425 


10-055993 


-124318 


185;9-875682 


141 


20 


9-819832 


239 


•1801681 9-944262; 425 


10-055738 


-124429 


1859-875571 


i40 


21 


9-819976 


239 


•180024 li 9-9445171 425 


10-055483 


-124541 


1859-875459 


39 


22 


9-820120 


239 


•179880 P 9-944771 425 


10-055229 


-124652 


185 9-875348 


138 


23 


9-820263 


239 


-179737 1' 9-9450261 425 


10-054974 


-124763 


l85;9-875237i;37 | 


24 j 


9-820406 


239 1 


•179594 9-94-52811 425 !10-0-54719 


-124874 


185:9-875126 


36 


25 1 


9-820550 


239; 


•179450 9-945535 425 


10-054465 


-124986 


186:9-875014 


,35 


26 


9-820693 


238 


•179307 9-945790| 425 


10-0-54210 


-125097 


186:9-874903 


!34 


27 


9-820836 


238 


•179164 9-946045;425 


10 053955 


-125209 


18619-874791 


:33 


28 


9-820979 


238 


•179021 9-946299:425 


10-053701 


•125320 l86!9-874680'32 


29' 


9-821122 


238 


•178878 9-946554 425 


10-053446 


•125432;i86!9-874568'l31 


30 


9-821265 


238 


•178735 .9-9468081425 


10-053192 


-1255441186. 9-874456:30 


31 


9-821407 


238 


•178593 '1 9-947063' 425 110-052937 


-12-5656 


186 9-874344 29 


32 


9-821550 


238 


•178450 ' 9-947318' 424 10-052682 


-125768 


l86i9-874232'28 


33 


9-821693 


238 


•178307 -9-947572 424 10-052428 


•125879 


186|9-874121'27 


34 


9-821835 


237 


•178165 ii 9-947826 424 aO-052174 


•125991 


187 


9-874009 '26 


35 


9-821977 


237 


•178023 : 9-948081 424 110-051919 


•126104 


187 


9-873896 25 


36 


9-822120 


237 


•177880 ■ 9-948336; 424 10-051664 


•126216 


187 


9-873784'24 


37 


9-822262 


237 


•177738 9-9485901424:10-051410 


-126328 


187 


9-873672 23 


38 


9-822404 


237 


•177596 i; 9-948844 


424 ,10-051156 


•1264401187 


9-873560 22 


39 


9-822546 


237 


•177454 19-949099 


424 '10-0-50901 


•126552187 


9-873448 21 


40 


9-822688 


237 


•177312 19-949353 


424 ;i0050647 


•126665187 


9-87333520 


41 


9-822830 


236 


•177170 9-949607 


424 '10050393 


■126777187 


9-873223,19 : 


i 42 


19-822972 


236 


•177028': 9-949862 


424 !l0-0.50138 


•126890;187 


9-873110 18 


43 


9-823114 


236 


•176886 ij 9-950116 


424 110-049884 


•127002'l88 


9-872998 17 


44 


9-823255 


236 


•176745 ii 9-950370 


424 jlO-049630 


•127115:188 


9-872885 16 


45 


9-823397 


236 


■176603 '9-950625 


424 '10-049375 


•127228:188 


9-872772 15 


46 


9-823539 


236 


•176461 i 9-950879i 424 10-049121 


•127341il88 


9-872659 14 | 


47 


9-823680 


236 


•176320 


9-9511331 424 10-048867 


•127453:188 


9-872547 13 | 


48 


9-823821 


235 


•176179 j 


9-951388 


424 ;i0-048612 


•127566188 


9-872434 12 


! 49 


9-823963 


235 


-176037 


9^951642 


424 '10-048358 


•127679188 


9-872321111 1 


50 


9-824104 


235 


•175896 


9-951896 


424 10-048104 


-1277921188 


9-872208 10 i 


51 


9-824245 


235 


•175755 


9-952150 


424 10-047850 


-127905188 


9-872095' 9 1 


52 


9-824386 


235 


•175614 


9-952405 


424 10-047595 


|-128019'189 


9-871981;: 8 ! 


53 


9-824527 


235 


•175473; 


9-952659 


424 110-047341 ;;-128l32 189 


9-871868, 7 l 


54 


9-824668 


235 


•175332 


9-952913 


424 10-047087 1-128245 189 


9-871755,1 6 


55 


9-824808 


234 


•175192: 


9-953167 


424 ;10-046833;| -128359 189 


9-87164l! 5 


56 


9-824949 


234 


•175051 1 


9-953421 


423 !l0-046579 1-128472 189 


9-871528 i 4 


57 


9-826090 


234 


•174910 


9-953675' 423 !l0-046325 j -128.586 189 


9-871414' 3 1 


58 


9-825230 


234 


•174770 


9^953929i 423 ;10-046071 ;^128699 189 


9-871301 ! 2 ! 


59 


9-825371 


234 


•174629 


9-954183 423 jlO-045817 ;-128813l89 


9-871187:i 1 


60 


9-825511 


234 


•174489 


9-954437! 423 ilO-045563 -128327 189 9^871073|| 


! Cosius. 


' ^ecallt. 1 Cotangent. ■ ! Tangent. Cosecant. | Sine. || ' 




















48 DES. 



LOGARITHMIC SINES, ETC. 



83 



42 


DEC. 




















^' 


Sine. 


Diff. 

lOU" 


Cosecant. 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


Diff. 
100" 


Cosine. 


, 1 
1 





9-825511 




•174489 


9-954437 




10-045563 


•128927 




9-871073"60 1 


1 


9-825651 


234 


•174349 


9-954691 


423 


10-045309 


•129040 


190 


9-870960 59 i 


2 


9-825*791 


233 


-174209 


9-954945 


423 


10-045055 


•129154 


190 


9-870846 58 i 


3 


9-825931 


233 


•174069 


9-955200 


423 


10-044800 


•129268 


190 


9-870732 57 1 


4 


9-826071 


233 


-173929 


9-955454 


423 


10-044546 


•129382 


190 


9-870618 56 '< 





9-826211 


233 


•173789 


9-955707 


423 


10-044293 


•129496 


190J9-870504 55 \ 


6 


9-826351 


233 


-173649 


9-955961 


423 


10-044039 


•129610 


190 9-870390 54 i 


7 


9-826491 


233 


•173509 


9-956215 


423 


10-043785 


•129724 


19019-870276 53 ', 


8 


9-826631 


233 


•173369 


9-956469 


423 10-043531 


■129839 


19019-870161 52 : 


9 


9-826770 


233 


•173230 


9-956723 


423 10-043277 


•129953 


190!9-870047 51 i 


10 


9-826910 


232 


•173090 


9-956977 


423 '10-043023 


•130067 


191 


9-869933 50 ! 


11 


9-827049 


232 


•172951 


9-957231 


423 


10-042769 


•130182 


191 


9-869818 49 


12 


9-827189 


232 


-172811 


9-957485 


423 


10-042515 


•130296 


191 


9-869704 48 . 


13 


9-827328 


232 


•172672 


9-957739 


423 


10-042261 


•130411 


191 


9-869589,|47 1 


14 


9-827467 


232 


-172533 


9-957993 


423 


10-042007 


•130526 


191 


9-869474 46 ! 


15 


9-827606 


232 


•172394 


9-958246 


423 


10-041754 


•130640 


191 


9-869360 45 \ 


16 


9-827745 


232 


•172255 


9-958500 


423 


10-041500 


•130755 


191 


9-869245!44 j 


17 


9-827884 


232 


-172116 


9-958754 


423 


10-041246 


•130870 


191 


9-869130 43 i 


18 


9-828023 


231 


•171977 


9-959008 


423 


10-040992 


•130985 


19119-869015 42 ! 


19 


9-828162 


231 


•171838 


9-959262 


423 


10-040738 


•131100 19219-868900 141 \ 


20 


9-828301 


231 


•171699 


9-959516 


423 


10-040484 


-131215 


192 


9-868785 140 1 


21 


9-828439 


231 


•171561 


9-959769 


423 


10-040231 


•131330 


192 


9-868670 39 ' 


22 


9-828578 


231 


•171422 


9-960023 


423 


10-039977 


•131445 


192 


9-868555 38 i 


23 


9-828716 


231 


•171284 


9-960277 


423 


10-039723 


•131560 


192 


9-868440 37 


24 


9-828855 


231 


•171145 


9-960531 


423 


10-039469 


•131676 


192 


9-868324 36 : 


25 


9-828993 


230 


•171007 


9-960784 


423 


10-039216 


•131791 


192 


9-868209:35 


26 


9-829131 


230 


•170869 


9-961038 


423 


10-038962 


•131907 


192 


9-868093; 34 1 


27 


9-829269 


230 


•170731 


9-961291 


423 


10-038709 


•132022 


192 


9-867978 33 ! 


28 


9-829407 


230 


•170593 


9^961545 


423 


10-038455 


•132138 


193 


9-867862 32 | 


29 


9-829545 


230 


•170455 


9-961799 


423 


10-038201 


•132253 


193 


9-867747i31 


30 ' 


9-829683 


230 


-170317 


9-962052 


423 


10-037948 


•132369 


193 


9-867631 


30 


31' 


9-829821 


230 


-170179 


9-962306 


423 


10-037694 


•132485 


193 


9-867515 


29 


32 


9-829959 


229 


•170041 


9-962560 


423 


10-037440 


•132601 


193 


9-807399 


28 


33 


9-830097 


229 


•169903 


9-962813 


423 


10-037187 


•132717 


193 


9-867283 


27 


34 :j 9-8302341229 


•169766 


9-963067 


423 


10-036933 


•132833 


193 


9-867167 


26 


35 


9-830372 


229 


•169628 


9-963320 


423 


10-036680 


•132949 


193 


9-867051 


25 


36 


9-830509 


229 


•169491 


9-963574 


423 


10-036426 


•133065 


193 


9-866935 


24 


37 


9-830646 


229 


•169354 


9-963827 


423 


10-036173 


•133181 


194 


9-866819 


23 


38 


9-830784 


229 


•169216 


9-964081 


423 


10-035919 


•133297 


194 


9-866703 


22 


39 


9-830921 


229 


•169079 


9-964335 


423 


10-035665 


•133414 


194 


9-866586 


21 ! 


40 


9-831058 


228 


•168942 


9-964588 


423 


10-035412 


•133530 


194 


9-866470 


20 1 


41 


9-831195 


228 


•168805 


9-964842 


423 


10-035158 


•133647 


194 


9-866353 


19 


42 


9-831332 


228 


•168668 


9-965095 


422 


10-034905 


•133763 


194 


9-866237 


18 


43 


9-831469 


228 


•168531 


9-9653^9 


422 


10-034651 


•133880 


194 


9-866120 


17 


44, 


9-831606 


228 


•168394 


9-965602 


422 


10-034398 


•133996 


194 


9-866004 


16 


45 


9-831742 


228 


•168258 


9-965855 


422 


10-034145 


•134113 


195 


9-865887 


15 1 


46 


9-831879 


228 


•168121 


9-966109 


422 


10-033891 


•134230 


195 


9-865770 


14 ' 


47 |1 9-832015 


228 


•167985 


9-966362 


422 


10-033638 


-134347 


195 


9-865653 


13 \ 


48 


9-832152 


227 


•167848 


9-966616 


422 


10-033384 


•134464 


195 


9-865536 


12 1 


49 


9-832288 


227 


•167712 


9-966869 


422 


10-033131 


•134581 


195 


9-865419 


11 


50 


9-832425 


227 


•167575 


9-967123 


422 


10-032877 


•134698 


195 


9-865302 


10 


51 


9-832561 


227 


•167439 


9-967376 


422 


10-032624 


•134815 


195 


9-865186 


9 


52 


9-832697 


227 


•167303 


9-967629 


422 


10-032371 


-134932 


195 


9-865068 


8 


63 


9-832833 


227 


•167167 


9-967883 


422 


10-032117 


•135050 


195 


9-864950 


7 


54 


9-832969 


227 


•167031 


9-968136 


422 


10-031864 


-135167 


195 


9-864833 


6 


55 


9-833105 


226 


•166895 


9-968389 


422 


10-031611 


•135284 


196 


9-864716 


5 


56; 


9-833241 


226 


•166759 


9-968643 


422 


10-031357 


•135402 


196 


9-864698 


4 


57 i 


9-833377 


226 


•166623 


9-968896 


422 


10-031104 


•135519 


196 


9-864481 


3 


58: 


9-833512 


226 


•166488 


9-969149 


422 


10-030851 


•135637 196 


9-864363 


2 


59! 


9-833648 


226 


•166352 


9-969403 


422 


10-030597 


•135755 


196 


9-864245 


1 


60 1 


9-833783 


226 


•166217 1 


9-969656 


422 10-030344 


•135873 


196 


9-864127 





' 


Cosine. 


Secant. Cotangent. 


1 Tangent. |i Cosecant. 




Sine. 


"~ 



47 DJEG. 



84 



LOGARITHMIC SINES, ETC. 



43 DEG. 























Sine. 


Difif. 

100" 


Cosecant. 


1 
Tangent. 


Diff. 
100" 


Cotangent. 


Secant. 


Diff. 

100" 


Cosine. 


' 


9-833783 




•166217 


9-969656 




10-030344 


•135873 




9^864127 


60' 


111 9-833919 


226 


•166081 


9-969909 


422 


10-030091 


•135990196 


9-864010 !59 


2 1 9-834054 


225 


•165946 


9-970162 


422 


10-029838 


•136108196 


9-863892 ..58 


3 9-834189 


225 


•165811 


9-970416 


422 


10-029584 


-136226'l97 


9-863774157 


4 1 9-834325 


225 


•165675 


9-970669 


422 


10-029331 -136344197 


9-863656156 


5 


9-834460 


225 


-165540 


9-970922 


422 


10-029078 -1364621197 


9-863538 55 


6 


9-834595 


225 


•165405 


9-971175 


422 


10^028825 


•136581 197| 


9-863419 54 


7 


9-834730 


225 


•165270 


9-971429 


422 


10-028571 


•136699 


197 


9-863301 153 


8 


9-834865 


225 


•165135 


9-971682 


422 


10-028318 


-136817 


197 


9-863183 52 


9119-834999 


225 


•1-65001 


9-971935 


422 


10-028065 


-136936 


197 


9-863064 51 


I 10 9-835134 


224 


•164866 


9-972188 


422 


10-027812 


•137054 


197 


9-862946 '50 


i 11 ij 9-835269 


224 


•164731 


9-972441 


422 


10-027559 -137173 


198 


9-862827 49 


12 i 9-835403 


224 


•164597 


9-972694 


422 


10-027306-137291 


198 


9-862709 ^48 


13 i 9-835538 


224 


•164462 


9-972948 


422 


10-027052-137410 


198 


9 -862590; 47 


14 1 9-835672 


224 


•164328 


9-973201 


422 


10-026799 I -137-529 


198 


9^862471'j46 


15 J 9-835807 


224 


•164193 


9-973454 


422 


10-026546 '-137647 


198 


9^862353:k5 


16 1 


9-835941 


224 


•164059 


9-973707 


422 


10-026293 1 -137766 


198 


9^862234'44 


1' 


9-836075 


224 


•163925 


9-973960 


422 


10-026040 -137885 


198 


9^862115143 


18! 


9-836209 


223 


•163791 


9-974213 


422 


10-025787 i -138004 


198 


9-861996:142 


19 9-836343 


223 


•163657 


9-974466 


422 


10-025534 -138123 


198 


9-86l877|41 


20 ;i 9-836477 


223 


•163523 


9-974719 


422 110-0252811-138242 


198 


9-86175840 


21 j 9-836611 


223 


•163389 


9-974973 


422 I10-025027 -138362 


199 


9-861638 '39 


22 9-836745 


223 


•163255 


9-975226 


422 


10-024774: -138481 


199 


9-861519 38 


23 


9-836878 


223 


•163122 


9-975479 


422 


10-024521-138600 


199 


9-861400 37 


24 


9-837012 


223 


•162988 


9-975732 


422 


10-024268 'i-138720 


199 9-861280 '36 
199 9-861161 :35 


25 


9-837146 


222 


•162854 


9-975985 


422 


10-024015 -138839 


26 1 


9-837279 


222 


•162721 


9-976238 


422 


10-023762 -138959 


199 9^861041: 34 


27! 


9-837412 


222 


•162588 


9-976491 


422 


10-023509 -139078 


199 9-860922 '33 


28 1:9-837546 


222 


•162454 


9-976744 


422 


10-023256 -139198 


199 9-860802 32 


29 19-837679 


222 


•162321 


9-976997 


422 


10-023003-139318 


19919-860682 31 


i 30 


9-837812 


222 


•162188 


9-977250 


422 


10-022750 •139438l200!9-860562 30 | 


1 31 


9-837945 


222 


•162055 


9^977503 


422 


10-022497 


-1395581200:9-860442 29 


32 


9-838078 


222 


•161922 


9-977756 


422 


10-022244 


-139678'200J9-860322 28 


33; 9-8382111 


221 


•161789 


9-978009 


422 


10-021991 


•1397981200|9-860202 27 


34 


9-838344 


221 


•161656 


9-978262 


422 


10-021738 


-139918J200 9-860082 26 


35 


9-838477 


221 


•161523 


9-978515 


422 


10-021485 


-l40038'200l9-859962 125 


36 


9-838610 


221 


•161390 


9-978768 


422 


10-021232 


•140158i200 


9-859842124 


37 


9-838742 


221 


•161258 


9-979021 


422 110-020979 


•1402791200 


9-859721 123 


38 


9-838875 


221 


•161125 


9-979274 


422 10-020726 


-1403991201 


9-85960122 


1 39 


9-839007 


221 


•160993 


9-979527 


422 10-020473 


•140520 


201 


9-859480 21 


! 40 


9-839140 


221 


•160860 


9-979780 


422 i 10-020220 


' -140640 


201 


9-859360 20 


i 41 


9-839272 


220 


•160728 


9-980033 


422 


10-019967 


:-140761J201 


9-859239119 


i 42 


,9-839404 


220 


•160596 


9-980286 


422 


10-019714 


[•1408811201 


9-8-59119 18 


43 


19-839536 


220 


•160464 


9-980538 


422 


10-019462 


'•1410021201 


9-858998 17 


1 44 


9-839668 


220 


•160332 


9-980791 


422 


10-019209 


-141123201 


9-858877 16 


1 45 


9-839800 220 


•160200 


9-981044 


422 


10-018956 


-141244;201 


9-8-58756 15 


46 


9-839932 220 


•160068 


9-981297 


422 


10-018703 


-141365202 


9-858635!l4 


47 


9-840064 220 


•159936 


9-981550 


422 


10-018450 


i •1414861202 


9-858514 


[13 


48 


9-840196 219 


•159804 


9-981803 


422 


10-018197 


•141607 


202 


9-858393 


'l2 


49 


9-840328' 219 


•159672 


9-982056 


422 


10-017944 


-141728 


202 


9-858272 


11 


50 


9-8404591 219 


•159541 


9-982309 


422 


10-017691 


1-141849 


202 


9-858151 


10 


51 


[9-840591 


219 


•159409 


9-982562 


421 


10-017438 


-141971 


202 


9-858029 


9 


52 


9-840722 


219 


•159278 


9-982814 


421 


10-017186 


-142092 


202 


9-857908 


8 


53 


9-840854 


219 


•159146 


; 9-983067 


421 


10-016933 


•142214 


20219-857786 


7 


54 


9-840985 


219 


•159015 


! 9-983320 


421 


10-016680 


-142335 


2029-857665 


6 


55 


,9-841116 


219 


•158884 


9-983573 


421 


10-016427 


•142457 


20319-857543 


5 


56 


i 9-8412471 218 


•158753 i 9-983826 


421 


10-016174 


-142578J203{9-857422 
-142700 203|9-857300- 


4 


57 


9-8413781218 


•158622 


9-984079 


421 


10-015921 


3 


58 


9-841509! 218 


•158491 


9-984331 


421 110-015669 


-142822 20319-857178 
-142944120319-857056, 


2 


59 


1 9-841640 218 


•158360 


9-984584 


421 10-015416 


1 


60 


9-841771:218 


1 ^158229 


9-984837 421 :10-015163 


•143066,203j9-856934 







Cosine. 


1 Secant. Cotangent. ' t Tangent. 


'Cosecant. 1 | Sine. 1 



46 DE6. 



LOGARITHMIC SINES, ETC. 



85 



44 


DEC. 






















Siue. 


W0'> Cosecant. 


Tangent. 


Diff. 

100" 


Cotangent. 


Secant. 


Diff 
100" 


Cosine. 


1' 







9-«41771 


1 


•158229 


9-984837 




10-015163 


•143066 




9-856934 


160 


1 


9-841902 


218 1 


•158098 


9-985090 


421 


10-014910 


-143188 


203 


9-856812 


159 


2 


9-842033 


218 


-157967 


9-985343 


421 


10-014657 


-143310 


203 


9-856690158 


3 


9-842163 


218. 


•157837 


9-985596 


421 


10-014404 


•143432 


204 


9-8565681157 


4 


9-842294 


217 1 


•157706 


9-985848 


421 


10-014152 


-143554 


204 


9-856446! '56 


5 


9-842424 


217 1 


•157576 


9-986101 


421 


10-013899 


-143677 


204 


9-856323 ;55 


6 


9-842555 


217! 


•157445 


9-986354 


421 


10-013646 


-143799 


204 


9-856201 


54 


7 


9-842685 


217 


•157315 


9-986607 


421 


10-013393 


-143922 


204 


9-856078 


53 


8 


9-842815 


217 


•167185 1 


9-986860 


421 


10-013140 


-144044 


204 


9-855956 


52 


9 


9-842946 


217 


•157054 


9-987112 


421 


10-012888 


-144167 


204 


9-855833 


51 


10 


9-843076 


217 


•156924 


9-987365 


421 


10-012635 


-144289 


204 


9-855711 


50 


11 


9-843206 


217 


•156794 1 


9-987618 


421 


10-012382 


-144412 


205 


9-855588 


49 


12 


9-843336 


216 


•1566641 


9-987871 


421 


10-012129 


-144535 


205 


9-855465 


48 


13 


9-843466 


216 


-1565341! 9-988123 


421 


10-011877 


•144658 


205 1 9 -8-55342 


147 


14 


9-843595 


216 


•156405 1 9-988376 


421 


10-011624 


-144781 


20519-855212 


j46 


15 


9-843725 


216 


•156275 1 


9-988629 


421 


10-011371 


-144904 


205 


9-855096 


145 


16 


9-843855 


216 


•156145 i 


9-988882 


421 


10-011118 


•145027 


205 


9-854973 


44 


17 


9-843984 


216 


•156016 1 


9-989134 


421 


10-010866, 


•145150 


205 


9-854850 


43 


18 


9-844114 


216 


•155886 ' 


9-989387 


421 


10-010613 


-145273 


205 


9-854727 


142 


19 


9-844243 


216 


•155757 ' 


9-989640 


421 


10-010360 


•145397 


206 


9-854603 


'41 


20 


9-844372 


215 


•155628 


9-989893 


421 


10-010107 


•145520 


206 


9-854480 


40 


21 


9-844502 


215 


-155498 


9-990145 


421 


10-009855 


•145644 


206 


9-8-54356 '39 


22 


9-844631 


215 


•155369 


9-990398 


421 


10-009602 


-145767 


206 


9-8542331138 


231! 9-844760 


215 


•155240 ! 


9-990651 


421 


10-009349 


-145891 


206 


9-854109 137 


24!! 9-844889 


215 


•155111 ' 


9-990903 


421 


10-009097 


-146014 


206 


9-853986 


:36 


25 


9-845018 


215 


•154982 


9-991156 


421 


10-008844 


-146138 


206 


9^853862 


i35 


26 


9-845147 


215 


•154853 


9-991409 


421 


10-008591 


-146262 


206 


9-853738 


34 


27 


9-845276 


215 


•154724^1 9-991662 


421 


10-008338 


-146386 


206 


9-853614ll33 


28 |! 9-845405 


214 


•154595 11 9-991914! 421 
•154467 9-992167:421 


10-008086 


■146510 


207 


9-853490"32 


29119-845533 


214 


10-007833 


-146634 


207 


9-853366'31 


30 


9-845662 


214 


•1543381 9-992420 


421 


10-007580 


-146758 


207 


9-853242 30 
9-853118129 


31 


9-845790 


214 


-1542101' 9-992672 


421 


10-007328 


•146882 


207 


32 


9-845919 


214 


•154081 


9-992925 


421 


10-007075 


-147006 


207 


9-85299428 


33 


9-846047 


214 


•153953 


9-993178 421 


10-006822 


-147131 


207 


9-852869 27 


341:9-846175 


214 


-153825 


9-993430 421 


10-006570 


•147255 


207 


9-852745126 


35 ! 9-846304 


214 


•153696 


9-993683 


421 


10-006317 


•147380 


207 


9-852620l'25 


36 


1 9-846432 


214 


•153568 


9-993936 


421 


10-006064 


-147504 


207 


9-852496l'24 


37 


19-846560 


213 


•153440 


9-994189 


421 


10-005811 


-147629 


208 


9-852371123 
9-8522471I22 


38 j 9-846688 


213 


-153312 


9-994441 


421 


10-005559 


-147753 


208 


39119-846816 


213 


•153184 


9-994694 


421 


10-005306 


-147878 


208 


9-852122i|21 


40119-846944 


213 


-153056 


9-994947 


421 


10-005053 


-148003 


208 


9-851997 i20 


41 il 9-847071 


213 


-152929 


9-99-5199 


421 


10-004801 


-148128 


208 


9-851872:119 


42 Ij 9-847199 


213 


•152801 


9-995452 


421 


10-004548 


-148263 


208 


9-851747118 


43 i 9-847327 


213 


-152673 


9-995705 


421 


10-004295 


-148378 


208 


9-8516221117 


44 


9-847454 


213 


•152546 


9-995957 


421 


10-004043 


-148503 


208!9-851497lil6 1 


45 


1 9-847582 


212 


-152418 


9-996210| 421 


10-003790 


-148628 


209!9-851372!jl5 1 


46 


19-847709 


212 


•152291 


9-9964631 421 


10-003537 


-148764 


2099-851246 


14 


47 


9-847836 


212 


•152164 


9-996715 421 


10-003285 


•148879 


2099-851121 


13 


48 


1 9-847964 


212 


•152036 


9-996968I 421 


10-003032 


-149004 


2099-850996 


12 


49 


9-848091 


212 


•151909 


9-997221 


421 


10-002779 


•149130 


20919-850870 


11 


50 


9-848218 


212 


•151782 


9-997473 


421 


10-002527 


•149255 


209 9-850745 


10 


51 


9-848345 


212 


-151655 


9-997726 


421 


10-002274 


•149381 


2099-850619 


9 


52 


9-848472 


212 


-151528 


9-997979 


421 


10-002021 


-149507 


209 9-850493 


8 1 


53 


9-848599 


211 


-151401 


9-998231 


421 


10-001769 


-149632 


210|9-850368 


■7 


54 


19-848726 


211 


•151274 


9-998484 


421 


10-001516 


-149758 


21019-850242 


G 


55 "9-848852 


211 


-151148 


9-998737 


421 


10-001263 


-149884 


2109-850116 


5 


56 '9-848979 


211 


-151021 


9-998989 


421 


10-001011 


-150010 


2109-849990 


4 


57 9-849106 


211 


-150894 


9-999242 


421 


10-000758 


-150136 


2109-849864 


3 


58 ;1 9-849232 211 


-150768 


9-999495 


421 


10-000505 


-150262 


210i9-849738 


2 


59 


9-849359 211 


•150641 


9-999747 


421 


10-000253 


-150389 


2109-849611 


1 1 


60 


9-849485 211 


•150515 


10-000000 


421 


10-000000 


-150515 21019-849485 


!0 


' ' Cosine. Secant. ' Cotangent. 




Tangent. ' Cosecant. ! Sine. 


i ' 



45 DEQ. 



INDEX, 



ABBREviATiONof the reduction of decimals, 17. 

Abrasion, limits of, 301. 

Absolute resistances, 288. 

Absolute strength of cylindrical columns, 274. 

Accelerated motion, 386. 

Accelerated motion of wheel and axle, 419. 

Acceleration, 415. 

Acceleration and mass, 422. 

Actual and nominal horse power, 240. 

Addition of decimals, 22. 

Addition of fractions, 20. 

Adhesion, 297. 

Air, expansion of, by heat, 173. 

Air that passes through the fire for each horse 

power of the engine, 210. 
Air, water, and mercury, 355. 
Air-pump, 254. 
Air-pump, diameter of, eye of air-pump cross 

head, 145. 
Air-pump machinery, dimensions of several 

parts of, 144. 
Air-pump strap at and below cutter, 147. 
Air-pump studs, 144. 
Ale and beer measure, 8. 
Algebra and arithmetic, characters used in, 12. 
Algebraic quantities, 134. 
Alloys, strength of, 287. 
Ambiguous cases in spherical trigonometry, 

381. 
Amount of effective power produced by steam, 

266. 
Anchor rings, 90. 
Angle iron, 91, 408, 409, 410. 
Angles of windmill sails, 445. 
Angles, measurement of, by compasses only, 

382. ' 
Angular magnitudes, 359. 
Angular magnitudes, how measured, 373. 
Angular velocity, 412. 
Apothecaries' weight, 6. 
Apparent motion of the stars, 353. 
Application of logarithms, 334. 
.Approximating rule to find the area of a seg- 
ment of a circle, 67. 
Approximations for facilitating calculations, 

00. 

Arc of a circle, to find, 49. 

Arc of one minute, to find the length of, 361. 

Arc, the length of which is equal to the ra- 
dius, 357. 

Architecture, naval, 453. 

Arcs, circulnr, to find the lengths of, 68. 

Area of segment and sector of a circle, 51. 

Area of steam passages, 220. 

Areas of circles, 57. 

Areas of segments and zones of circles, 64, 
65, 66, 67. 

Arithmetic, 10. 

Arithmetical progression, to find the square 
root of numbers in, 126. 

Arithmetical solution of plane triangles, 366. 



Arithmetical proportion and progression, 35 

to 38. 
Ascent of smoke and heated air in chimneys, 

208. 
Atmospheres, elastic force of steam in, 195, 

196. 
Atmospheric air, weight of, 356. 
Average specific gravity of timber, 396. 
Avoirdupois weight, 6. 
Axle and wheel, 417. 
Axle of locomotive engine, 168, 169. 
Axle-ends or gudgeons, 301. 
Axles, friction of, 298, 300. 

Balls of cast iron, 407. 

Bands, ropes, &c., 267. 

Bar iron, 400. 

Beam, 151. 

Beam, the strongest, 276. 

Bearings of water wheels, 285. 

Beafings or journals for shafts of various 
diameters, 287. 

Beaters of threshing machine, 445. 

Before and behind the piston, 232. 

Blast pipe, 171. 

Blistered steel, 281. 

Blocks, cords, ropes, shelves, 428. 

Bodies, cohesive power of, 175. 

Bodies moving in fluids, 324. 

Boiler, 171. 

Boiler plate, experiments on, at high tempe- 
ratures, 220, 

Boiler plates, 403. 

Boilers, 256 and 257. 

Boilers of copper and iron, diminution of 
the strength of, 219. 

Boilers, properties of, 215. 

Boilers, strength of, 218. 

Bolts and nuts, 406. 

Bolts, screw and rivet, 220. 

Boring iron, 445. - 

Bossut and Michelloti, experiments on the 
discharge of water, 319. 

Boyle of Cork, 200. 

Bramah's press, 427. 

Branch steam-pipe, 148. 

Brass, copper, iron, properties of, 280. 

Brass, round and square, 408. 

Breast wheels, 328. 

Breast and overshot wheels, maximum ve- 
locity of, 443. 

Buckets and shrouding of water wheels, 446. 

Building, to support with cast iron columns, 
293. 

Bushel, 5. 

Butt for air-pump, 146. 

Butt, thickness and breadth of, 14^. 

Butt, to find the breadth of, 141. 

Byrne's logarithmic discovery, 340. 

Byrne's theory of the strength of materials, 
272. 

583 



584 



INDEX. 



Calculation in the art of sMp-building, 453 
to 494. 

Calculation of Friction, 267. 

Carriages, motion of, on inclined planes, 429. 

Carriages travelling on ordinary roads, 307. 

Carrier or intermediate wheels, 434. 

Carts on ordinary roads, 311. 

Cases in plane trigonometry, 363. 

Cast iron, 174. 

Cast iron pipes, 404. 

Centre of effort, 483. 

Centre of gravity, 175. 

Centre of gravity of displacement of a ship, 
456, 457, 458. 

Centre of gyration, 180. 

Centre of oscillation, 187. 

Centres of bodies, 386. 

Centres of gravity, gyration, percussion os- 
cillation, 391. 

Centripetal and centrifugal forces, 178, 450. 

Chain bridge, 412. 

Chimney, 171, 208, 257. 

Chimney, size of, 212. 

Chimney, to -what height it may be carried 
with safety, 212. 

Circle, calculations respecting, 48, 49, 50, 53. 

Circle of gyration in water wheels, 444. 

Circles, 57 to 61. 

Circles, areas of, 57 to 63. 

Circular arcs, 68. 

Circular motion, 422. 

Circular parts of spherical triangles, 375. 

Circumference of a circle to radius 1, 361. 

Circumferences of circles, 57. 

Cloth measure, 7. 

Coefficient of efflux, 314. 

Coefficients of friction, 299. 

Cohesive strength of bodie-s, how to find, 281. 

Collision of railway trains, 452. 

Columns, eomparati\'e strength of, 294. 

Combinations of algebraic quantities, 134. 

Common fractions, 15. 

Common materials, 280. 

Complementary and supplementary ares, 374. 

Compound proportion, 14. 

Condenser, 226. 

Condensing water, 223. 

Conduit pipes, discharge by, 322. 

Cone, 82. 

Conical pendulum, 185 to 187. 

Connecting rod, 140, 141, 253. 

Continuous circular motion, 432. 

Contraction by efflux, 316. 

Contraction of the fluid vein, 313. 

Contractions in the calculation of loga- 
rithms, 348. 

Copper boilers, 219. 

Copper, iron, and lead, 405. 

Cosine, to find, 361. 

Cosines, contangents, &c., for every degree 
and minute in the quadrant, 540 to 576. 

Cosines, natural, 411. 

Cover on the exhausting side of the valve, 
in parts of the length of stroke, 231. 

Cover on the steam side, 226, 

Crane, 427. 

Crane, sustaining weight of, 285, 

Crank at paddle centre, 135. 

Crank axle, diameter of the outside bearings 
of, 168. 



Crank axle of locomotive, 169. 

Crank pin, 170, 252. 

Crank pin journal, 252. 

Crank pin journal, to find the diameter of, 139 

Crank pin journal, to find the length of, 139. 

Cross head, 252. 

Cross head, to find the breadth of eye of, 139 

Cross head, to find the depth of eye of, 139 

Cross multiphcation, 27. 

Cross tail, 253. 

Cube, 79. 

Cube and cube roots of numbers, 100 to 116. 

Cube root of numbers containing decimals, 
128. 

Cube root, to extract, 32. 

Cubes, 397 to 400. 

Cubes, to extend the table of, 128. 

Curve, to find the length of, by construction,72. 

Curves, to find the areas of, 453. 

Cuttings and embankments, 97. 

Cylinder side rods at ends, to find the diame- 
ter of, 143. 

Cylinders, 80, 397 to 400. 

Cylinders of cast iron, 404. 

Dams inclined to the horizon, 316. 

Decimal approximations for facilitating cal- 
culations, 55. 

Decimal equivalents, 56. 

Decimal fractions, 22. 

Decimal fractions, table of, 73. 

Decimals, addition of, 22. 

Decimals, division of, 24. 

Decimals, multiplication of, 23. 

Decimals, reduction of, 25, 26. 

Decimals, rule of three in, 27. 

Decimals, subtraction of, 23. 

Deflection of beams, 295, 

Deflection of rectangular beams, 294. 

Depth of web at the centre of main beam, 150. 

Destructive efl"ects produced by carriages on 
roads, 311. 

Devlin's oil, 297. 

Diagram of a curve of sectional areas, 460. 

Diagram of indicator, 265. 

Diameter of cj^linder, 251. 

Diameter of main centre journal, 143. 

Diameter of plain part of crank axle, 169. 

Diameter of the outside bearings of the crank 
axle, 168. 

Diameters of wheels at their pitch circle to 
contain a required number of t«eth, 436. 

Dimensions of the several parts of furnaces 
and boilers, 254. 

'Direct method to calculate the logarithm of 
any number, 346. 

Direct strain, 278, 

Discharge by compound tubes, 321, 

Discharge by different apertures from differ- 
ent heads of water, 318. 

Discbarge of water, 446. 

Discharges from orifices, 426. 

Displacement of a ship when treated as a 
floating hodj, 455. 

Displacement of ships, by vertical and hori- 
zontal sections, 460, 494. 

Distance of the piston from the end of its 
stroke, when the exhausting port is shut 
and when it is open, 231. 

Distances, how to measure, 369. 



INDEX. 



585 



Dirision by logarithms, Z?>C). 

Dodecaedron, 89. 

Double acting engines, rods of. 250. 

Double position, 44. 

Double table of ordinate s, 457. 

Drainage of water through pipes, 325. 

Dr. Dalton, and his countrj'mau, Dr. Young, 

of Dublin, 
Drums, 422. 

Drums in continuous circular motion, 432. 
Dry or corn measure, 8. 
Duodecimals, 27. 
Dutch sails of windmills, 333. 
D. valves, 233. 
Dynamometer, used to measure force, 269. 

Edttction ports, 171. 

Effective discharge of water, 314. 

Effective heating surface of flue boilers, 256. 

Effects of carriages on ordinary roads, 311. 

Elastic force of steam, 188. 

Elastic fluids, 205. 

Elliptic arcs, 69, 70, 71, 72. 

Embankments and cuttings, 97. 

Endless screw, 431. 

Engineering and mechanical materials, 386. 

Engine, motion of steam in, 206. 

Engine tender tank, 92. 

Enlargements of pipes, interruption of dis- 
charge by, 321. 

Evolution, 29. 

Evolution by logarithms, 339. 

Eye, diameter of, 251. 

Eye of crank, 136. 

Eye of crank, to find the length and breadth 
of large and small, 142. 

Eye of round end of studs of lever, 143. 

Examples on the velocity of wheels, drums, 
and pulleys, 438. 

Exhaust port, 230. 

Expanded steam, 236. 

Expansion, 237. 

Expansion, economical effect of, 216. 

Experiments on the strength and other pro- 
perties of cast iron, 174. 

Explanation of characters, 12. 

Extended theory of angular magnitude, 374. 

Exterior diameter of large eye, 252. 

Extraction of roots by logarithms, 339. 

Fall of water, 444. 

Feed pipe, 150. 

Feed water, 222. 

Felloes of wheels, 309. 

Fellowship, or partnership, 41. 

Fire-grate, 171, 214. 

Fit/gerald, 264, 269. 

Flange, 91. 

Flat bar iron, 407. 

Fiat iron, 400, 

Flexure by vertical pressure, 292. 

Flexure of revolving shafts, pillars, &e., 296. 

Flues, 256. 

Flues, fires, and boilers, 217. 

Fluids, the motion of elastic, 205. 

Fluids, to find the specific gravity of, 392. 

Fluids, the pressure of, 448. 

Fluid vein, contraction of, 313. 

Foot-valve passage, 149. 

Force. 267. 

Force, loss of, in steam pipes, 221. 



{ Force of steam, 188. 

! Forces, centrifugal and centripetal, 178, 450. 
I Fore and after bodies of immersion, 456, 460. 
I Form, the strongest. 275. 

Formulas for the strength of various parts 
of marine engines, 251. 

Formulas to find the three angles of a sphe- 
rical triangle when the three sides are 
given, 385. 

Formula, verj'' useful, 271. 

Fourth and fifth power of numbers, 129. 

Fractions, common, 15. 

Fractions, reduction of, 16, 17, 18, 19. 

Fractions, addition of, 20. 

Fractions, subtraction of, 21. 

Fractions, multiplication of, 21. 

Fractions, division of, 21. 

Fractions, the rule of three in, 21. 

Fractions, decimal, 22. 

Fractions, table of, 73. 

Fractious, addition contracted, 78. 

Fracture. 292. 

Franklin Institute, 172, 219. 

French litre, 355. 

French measures, 5, 6. 

French metre, 347. 

Friction, 238. 

Friction, coefBcents of, 300. 

Friction of fluids, 325. 

Friction of rest and of motion, 267. 

Friction of steam engines of different modi- 
fications, 302. 

Friction ofwater against the sides of pii*s, 321. 

Friction of water-wheels, windmills, &c., 267. 

Friction, or resistance to motion, in bodies 
rolling or rubbing on each other, 297. 

Friction, laws of, 298. 

Frustums, 83. 

Frustum of spheroid, 87. 

Furnace, 256. 

Furnace room, 213. 

Gallon, 5. 
Gases, 394. 
Geering, 422. 

General and universal expression, 376. 
General observations on the steam engine, 259. 
General trigonometrical solutions, 365, 369. 
Geometrical construction, 362. 
Geometrical construction of the proportion 

of the radius of a wheel to its pitch, 440. 
Geometrical proportion and progression, 33. 
Gibs and cutter, 140, 253. 
Gibs and cutter through air pump cross-head, 

146, 147. 
Gibs and cutter through cross-tail and 

through butt, 141. 
Gibs and cutter, to find the thickness and 

breadth of, 143. 
Girder, 275. 

Girth, the mean in measuring, 94. 
Glenie, the mathematician, 287. 
Globe, 85. 
Grate surface, 255. 
Gravity, centre of, 175, 386. 
Gravity, specific, 391. 
Gravity, weight, mass, 386. 
Gudgeons, 420. 
Gyration, centre of, 180, 390. 
Gvration, the centre of different figures and 
"bodies, 181. 



586 



INDEX. 



Heads of water, 318. 

Heating surface, 256. 

Heating surface of boilers, 215. 

Heights and discharges of water, 319. 

Heights and distances, 359. 

Height of chimneys, 210. 

Height of metacentre, 489, 483. 

Hewn and sawed timber, 95. 

Hexagon, heptagon, 48. 

High pressure and condensing engines, 234. 

Hollow shafts, to find the strength of, 284. 

Horizontal distance of centre of radius bar, 

246, 247. 
Horse power, 240. 
Horse power of an engine, dimensions made 

to depend upon the nominal horse power 

of an engine, 147. 
Horse power of pumping engines, 447. 
Horse power, tables of, 243, 244. 
Hot blast, 174. 
Hot liquor pumps, 446. 
Hydraulic pressure working machinery, 330. 
Hydraulics, 267, 312. 
Hydrogen, weight of, 356. 
Hydrostatic press, 448. 
Hyperboloid, 88. 

Hyperbolic logarithms, 130 to 133. 
Hyperbolic logarithms, how to calculate, 353. 
Hypothenuse of a spherical triangle, to find, 

378. 
Hypothenuse, 47. 

ICOSAEDRON, 89. 

Immersed portions of a ship, to calculate, 
456. 

Immersion and emersion, 453 to 467. 

Impact, 449. 

Impinging of elastic and inelastic bodies, 
452. 

Inaccessible distances, 372. 

Inches in a solid foot, 96. 

Inclined plane, 428, 429, 430. 

Inclination of the traces of ordinary car- 
riages, 311. 

Inclinations, discharge of a 6-inch pipe at 
several, 326. 

Increase of efiiciency arising from working 
steam expansively, 262. 

Index of logarithms, 334. 

Indicator, 264, 265. 

Indicator, the amount of the efi'eetive power 
of steam by, 266. 

Induction ports, 171. 

Inelastic lx)dies, 449. 

Influence of pressure, velocity, width of fel- 
loes, and diameter of wheels, 309. 

Initial plane, 456, 480, 500. 

Initial velocity with a free descent, 388. 

Injection pipe, 150. 

Inside discharging turbine, 330. 

Integer, 10. 

Integers, to find the square root of, 125. 

Interest, simple, 42. 

Interest, compound, 43. 

Involution, 28. 

Involution, or the raising of powers by loga- 
rithms, 338. 

Irregular polygons, 54. 

Iron, forged and wrought, 272. 

Iron plates, 403. 

Iron, properties of, 175. 



Iron, strength of, 173. 

Iron, taper and parallel, angle and T, rail- 
way and sash, 408, 411. 

Jet, specific gravity of, 394. 

Journal of cross-head, to find diameter of,139. 

Journal of cross-head, to find the length' 

of, 139. 
Journal, the mean centre, to find the diameter 

of, 143. 
Journal, strain of, 252. 
Journals for air-pump cross-head, 145. 
Journals for shafts of various diameters, 287. 
Julian year, 357. 
Juste Byrge, the inventor of logarithms, 133. 

Kane, Fitzgerald, 269. 
Keel and keelson, 433 to 500. 
Kilometre, 5. 
Kilogramme, 6. 
Knots, nodes, &c., 412. 

Lathe spindle wheel, 435. 

Laying off of angles by compasses only, 384. 

Leg of a spherical triangle, to find, 377. 

Length of crank pin of locomotive, 170. 

Length of paddle-shaft journal, 138. 

Length of stroke, 227, 251. 

Lengths that may be given to stroke of the 

valve, 229. 
Lengths of circular arcs, 68. 
Lever, 426. 

Light displacement, 459. 
Line of dii-ection, 390. 
Link next the radius bar, 242. 
Living forces, or the principle of vis viva,270. 
Load immersion, 456, 457. 
Load-water line, 456, 478. 
Locomotive engine, parts ofthe cylinder, 171. 
Locomotive engine, diameter of the outside 

bearings for, 163. 
Locomotive engine, dimensions of several 

moving parts, 171. 
Locomotive engine, dimensions of several 

pipes, 171. 
Locomotive engine, parts of the boiler, 171. 
Locomotive engine, tender tank, 92. 
Locomotive and other engines, 233. 
Logarithmic calculations, 376. 
Logarithmic calculations of the force of 

steam, 190 to 193. 
Logarithmic sines, tangents, and secants for 

every minute in the quadrant, 540, 576. 
Logarithms applied to angular magnitudes, 

359. 
Logarithms, hyperbolic, 130. 
Logarithms of the natural numbers from 1 

to 100000 by the help of difi"erences, 503 

to 540. 
Logarithms, the application of, 334. 
Long measure, 7. 
Longitudinal distance ofthe centre of gravity 

of displacement, 470, 500. 
Loss of force by the decrease of temperature 

in the steam pipes, 221. 
Low pressure engines, 243. 
Lunes, 54. 

Machinery, elements of, 425. 
Machinery worked by hydraulic pressure, 
330. 



INDEX. 



587 



Major and minor diameters of cross-head, 

253. 
Main bea6i at centre, 249. 
Malleable iron, 396. 
Marble, 288. 
Marine boilers, 217. 
Mass, 267. 

Mass, gravity, and weight, 386. 
Mass of a body, to find, \Yhen the weight is 

given, 389. 
Materials employed in the construction of 

machines, 267. 
Materials, their properties, torsion, deflexion, 

&c., 267. 
Maximum accelerating force, 421. 
Maximum velocity and power of water 

wheels, 443. 
Measures and weights, 5. 
Measurement of angular magnitudes, 374. 
Measurement of angles by compasses only, 

382. 
Mechanical effect, 417. 
Mechanical powers, 422. 
Mechanical power of steam, 261. 
Mensuration of solids, 79. 
Mensuration of timber, 93. 
Mensuration of superficies, 45. 
Mercury, density of, 350. 
Mercurj-^, to calculate the force of steam in 

inches of, 201. 
Method to calculate the logarithm of any 

given number, 340. 
MeUcentre, 482. 
Metre, 5. 
Midship, or greatest transverse section, 460, 

487. 
Millboard, 405. 
Millstones, 445. 
Millstones, strength of, 451. 
Modulus of elasticity, 278. 
Modulus of logarithms, 343. 
Modulus of torsion and of rupture, 279. 
Moment of inertia, 412. 
Motion of elastic fluids, 205. 
Motion of steam in an engine, 206. 
Multiplication of decimals, 23. 
Multiplication of fractions, 21. 
Multiplication by logarithms, 335. 
Musical proportion, 40. 

Natural sines, cosines, tangents, cotangents, 

secants, and cosecants, to every degree of 

the quadrant, 411. 
Naval architecture, 453. 
New method of multiplication, 342. 
Nitrogen, weight of, 356. 
Nominal horse power, tables of, for high and 

low pressure engines, 243, 244. 
Notation and numeration, 10. 
Notation, trigonometrical, 359. 
Number corresponding to a given logarithm, 

351. 
Number of teeth, or the pitch of small 

wheels, 435. 
Numbers, fourth and fifth powers of, 129. 
Numbers, logarithms of, 540, 495. 
Numbers, reciprocals of, 73 to 78. 
Numbers, squares, cubes, &e., of, 100 to 116. 
Numeral solution of the several cases of 

trigonometry, 361. 
Nuts; and bolts, 406. 



Oak, Dantzic, 280. 

Obelisk, to find the height of, 371. 

Oblique triangles, 368. 

Observatory at Paris g = 9*80896 metres,346. 

O'Byrne's turbine tables, 331. 

Octagon, 48. 

Octaedron, 89. 

O'Neill's experiments, 447. 

O'Neill's rules employed in the art of ship- 
building, 454. 

Opium, specific gravity of, 394. ^ 

Orders of lever, 426. 

Ordinates employed in the art of ship-build- 
ing, 455, 456, 458. 

Orifices and tubes, discharge of water by, 312. 

Orifices, rectangular, 314, 

Oscillation, centre of, 187, 391. 

Outside bearings of crank axle, 168. 

Outside discharging turbines, 331. 

Overshot wheels, 329. 

Overshot wheels, maximum velocity of, 443. 

Ox-hide, 299. 

Oxygen, 214, 356. 

PADDLE-shaft journal, 137, 251. 

Paraboloid, 88. 

Parabolic conoid, 88. 

Parallel angle iron, 409. 

Parallel motion, 242 to 246. 

Parallelogram of forces, 422. 

Parallelopipedon, 80. 

Partnership, 41. 

Partial contraction of the fluid vein, 316. 

Passages, area of steam, 220. 

Peclet's expression for the velocity of smoke 

in chimneys, 213. 
Pendulums, 183, 391. 
Pendulum, conical, 184. 
Pendulums, vibrating seconds at the level of 

the sea in various latitudes, 393. 
Percussion, centre of, 391. 
Periodic time, 179. 

Permanent weight supported by beams, 284. 
Permutations and combinations, 44. 
Pillars, strength of, 293. 
Pinions and wheels in continuous circular 

motion, 432. 
Pipes, discharge and drainage of water 

through, 321, 322, 325. 
Pipes of cast iron, 395. 
Pipes for marine engines, 149. 
Piston, 251. 

Piston of steam engine, 414. 
Piston rod, 140, 171, 253. 
Piston rod of air-pump, 146. 
Pitch circle, 436. 
Pitch of teeth, 441. 
Pitch of wheels, 435, 439. 
Plane triangles, solution of, 364, 365. 
Plane trigonometry, 359. 
Planks, deals, 94. 
Polygons, 47, 48. 
Polygons, irregular, 54. 
Port, upper and lower, 229.i 
Position, double, 44. 
Position, single, 43. 
Pound, 5. 

Power, actual and nominal, 241. 
Power and properties of steam, 261. 
Power that a cast-iron wheel is capable of 

transmitting, 442. 



588 



INDEX. 



Power of shafts, 294. 

Practical application of the mechanical 
powers, 425, 

Practical limit to expansion, 261. 

Practical observations on steam engines, 260. 

Principle of virtual velocities, 423. 

Prism, 80. 

Prismoid, 85. 

Properties of bodies, 401, 

Proportional dimensions of nuts and bolts, 
406. 

Proportion, 14. 

Proportion, musical, 40. 

Proportion and progression, arithmetical, 35 
to 38. 

Proportion and progression, geometrical, 38 
to 40. 

Proportion, or the rule of three by loga- 
rithms, 338. 

Proportion of wheels for screw-cutting, 433. 

Proportions of boilers, grates, <fcc., 213. 

Proportions of the lengths of circular arcs, 68. 

Proportions of undershot wheels, 328. 

PuUeys, 422, 427. 

Pump and pumping engines, 446. 

Pumping engines, 422. 

Pyramid, 82. 

Pyrometer, 63. 

Quadrant, 359. 

Quadrant, log. sines, cosines, &c., for every 
minute in, 540, 576. 

Quadrant, natural sines and cosines for 
every degree of, 411. 

Quadrant, to take angles with, 370. 

Quantities, known and unknown, 134. 

Quantity of water that flows through a cir- 
cular orifice, 313, 319. 

Quiescence, friction of, 299. 

Radius bar, 242. 

Radius bar, length of, corrected, 248. 

Radius of the earth at Philadelphia, 356. 

Radius of gyration, 412. 

Radius, length of, in degrees, 357. 

Rails, temporary, 411. 

Railway carriage, 268. 

Railway iron, 410. 

Raising of powers by logarithms, 338. 

Reciprocals of numbers, 73 to 78. 

Recoil, 449. 

Rectangle, rhombus, rhomboides, to find 
the areas of, 45, 46. 

Reduction of fractions, 16, 17, to 19, 20. 

Regnault's experiments on oxygen, Ac, 356. 

Regular bodies, 90. 

Relative capacities of the two bodies under 
the same displacement, 456, 470. 

Relative strength of materials to resist tor- 
sion, 294. 

Revolving shaft, 250. 

Riga fir, 290. 

Right-angled spherical triangles, 374. 

Ring, circular, to find the area of, 53. 

Ring, cylindrical, 90. 

Roads, traction of carriages on, 307. 

Rolled iron, 395. 

Roman notation, 11. 

Rope, strength of, 282. 

Ropes, bands, &c., 267. 

Ropes, blocks, pulleys, 428. 



Ropes, stifiiiess of, resistance of, to bendin 

302. 
Ropes, tarred and dry, 304, 306. 
Rotative engines, 260. 
Rotation, moment of, 414. 
Rotation of a body about a fixed axis, 416. 
Rotations of millstones, 452. 
Round and rectangular bars, strength of, 281. 
Round bar-iron, 4U3. 
Round steel and brass, 408. 
Rules for pumping engines, 448. 
Rule of three, 13. 
Rule of three by logarithms, 338. 
Rule of three in fractions, 21. 
Rupture, 272. 

Safety valves, 149, 150, 224. 

SaUs of windmills, 332. 

Sash ii'on, 410. 

Scales of chords, how to construct, 360. 

Scale of displacement, 490. 

Scantling, 95. 

Screw cutting by lathe, 433. 

Screw, power of, 430. 

Screw, to, cut, 434. 

Sectional area measured, 456 to 468. 

Segments of circles, 64 to 67. 

Shelves, cords, blocks, 428. 

Ship-building and naval architecture, 453. 

Sidereal day, 9. 

Side lever, to find the depth across the centre 
of, 144. 

Side rod, 246, 254. 

Side rod of air-pump, 146. 

Sines, cosines, <fec., 411. 

Sines, tangents and secants, 359. 

Singular phenomena, 237. 

Sleigh, 268. 

Slide valve, 225. 

Slide valve, a cursory examination of, 232. 

Slopes 1^ to 1, 2 to 1, and 1 to 1, 97. 

Sluice board, 316. 

Smoke and heated air in chimneys, 202. 

Solid inches in a solid foot, 96. 

Solids, mensuration of, 79. 

Space described by a body during a free de- 
scent in vacuo, 388, 

Specific gravity, 386, 391. 

Sphere, 85. 

Spheres, 397 to 400. 

Spheroid, 86, 87, 88. 

Spherical trigonometry, 373. 

Spheroidal condition of water in boilers, 236. 

Spindle and screw wheels, 434. 

Square, to find the area of, 45. 

Square and sheet iron, 402. 

Squares and square roots of numbers, 100 
to 116. 

Square root, 30. 

Square root of fractions and mixed numbers, 
31. 

Square measure, 6. 

Stability, 459. 

Stars, apparent motion of, 353. 

Statical moment, 417. 

Steam engine, 135. 

Steam dome, 171. 

Steam passages, 220. 

Steam pipes, loss of force in, 222. 

Steam port, 147, 148. 

Steam room, 259. 



INDEX. 



589 



Steam, elastic force of, 188 to 202. 

Steam, temperature of, pressure of, 172. 

Steam, volume of, 202 to 206. 

Steam, weight of, 204. 

Steel, 408. 

Steel, cast, 409. 

Stiffness of ropes, 302, 306. 

Stowage, 503. 

Stowing the hold of a vessel, 453, 456. 

Strap at cutter, 141. 

Strap, mean thickness of, at and before cut- 
ter, 143. 

Strength of bodies, 282. 

Strength of boilers, 218. 

Strength of materials, 173, 271. 

Strength of rods when the strain is wholly 
tensile, 250. 

Strength of the teeth of cast iron wheels, 437. 

•" -'ds of lever, 143. 

l-wheel and pinion, 434. 

buotraction of decimals, 23. 

Subtraction of fractions, 21. 

Table by which to determine the number of 
teeth or pitch of small wheels, 435. 

Table containing the circumferences, squares, 
cubes, and areas of circles, from 1 to 100, 
advancing by a tenth, 57, 58, 59, 60 to 
63. 

Table containing the weight of columns of 
water, each one foot in length, in pounds 
avoirdupois, 401. 

Table containing the weight of square bar 
iron, 402. 

Table containing the surface and solidity of 
spheres, together with the edge of equal 
cubes, the length of equal cylinders, and 
weight of water in avoirdupois pounds, 
397. 

Table containing the weight of flat bar iron, 
400. 

Table containing the specific gravities and 
other properties of bodies ; water the stand- 
ard of comparison, 401. 

Table containing the weight of round bar 
iron, 403. 

Table containing the weights of cast iron 
pipes, 404. 

Table containing the weight of solid cylin- 
ders of cast iron, 404. 

Table containing the weight of a square foot 
of copper and lead, 405. 

Table for finding the weight of malleable 
iron, copper, and lead, 405. 

Table for finding the radius of a wheel when 
the pitch is given, or the pitch when the ra- 
dius is given, for any number of teeth, 439. 

Table for the general construction of tooth 
wheels, 442. 

Table for breast wheels, 329. 

Table of polygons, 48. 

Table of decimal approximations for facili- 
tating calculations, 55. 

Table of decimal equivalents, 56. 

Table of the areas of the segments and zones 
of a circle of which the diameter is unity, 
64, 65, 66, 67. 

Table of the proportions of the lengths of 
semi-elliptic arcs, 69, 70, 72. 

Table of flat or board measure, 93. 

Table of solid timber measure, 94. 



Table of reciprocals of numbers, or of the 
decimal fractions corresponding to com- 
mon fractions, 71 to 77, 78. 

Table of weights and values in decimal 
parts, 79. 

Table of regular bodies, 90. 

Table of the cohesive power of bodies, 175. 

Table of hyperbolic logarithms, 130 to 133. 

Table of the pressure of steam, in inches of 
mercury at different temperatures, 172. 

Table of the temperature of steam at differ- 
ent pressures, in atmospheres, 172. 

Table of the expansion of air by heat, 173. 

Table of the sti'ength of iron, 173. 

Table of the superficial and solid content of 
spheres, 96. 

Table of solid inches in a solid foot, 96. 

Table of squares, cubes, square and cube 
roots, of numbers, 100, 101, 116, 125. 

Table of cover on the exhausting side of the 
valve in parts of the stroke and distance 
of piston from the end of its stroke, 231. 

Table of the proportions of the lengths of 
circular arcs, 68. 

Table of the fourth and fifth power of num- 
bers, 129. 

Table of the properties of different boil- 
ers, 215. 

Table of the economical effects of expan- 
sion, 216. 

Table of the comparative evaporative power 
of different kinds of coal, 218. 

Table of the cohesive strength of iron boiler 
plate at different temperatures, 219. 

Table of diminution of strength of copper 
boilers, 219. 

Table of expanded steam, 239. 

Table of the proportionate length of bearings, 
or journals for shafts of various diameters, 
287. 

Table of tenacities, resistances to compres- 
sion and other properties of materials, 
288. 

Table of the strength of ropes and chains, 
288. 

Table of the strength of alloys, 289. 

Table of data of timber, 289. 

Table of the properties of steam, 261. 

Table of the mechanical properties of steam, 
263. 

Table of the cohesive strength of bodies, 281. 

Table of the strength of common bodies, 283. 

Table of torsion and twisting of common ma- 
terials, 286. 

Table of the length of circular arcs, radius 
being unity, 63. 

Table of experiments on iron boiler plate at 
high temperature, 220. 

Table of the absolute weight of cylindrical 
columns, 274. 

Table of flanges of girders, 276. 

Table of mean pressure of steam at different 
densities and rates of expansion, 239. 

Table of nominal horse power of high pres- 
sure engines, 244. 

Table of nominal horse power of low pres- 
sure engines, 243. 

Table of dimensions of cylindrical columns 
of cast iron to sustain a given load with 
safety, 293. 

Table of strength of columns, 294. 



590 



INDEX. 



Table of comparative torsion, 294. 

Table of the depths of square beams to sup- 
port from 1 cwt. to 14 tons, 295, 296. 

Table of the results of experiments on fric- 
tions, with unguents interposed, 299, 300. 

Table of the results of experiments on the 
gudgeons or axle-ends in motion upon their 
bearings, 301. 

Table of friction, continued to abrasion, 301. 

Table of friction of steam engines of differ- 
ent modifications, 302. 

Table of tarred ropes, 303. 

Table of white ropes, 305. 

Table of dry and tarred ropes, 306. 

Table of the pressure and traction of car- 
riages, 308. 

Table of traction of wheels, 309. 

Table of the ratio of traction to the load, 
310. 

Table of the coefficients of the efflux through 
rectangular orifices in a thin vertical plate, 
315. 

Table of the coefficients of efflux, 315. 

Table of comparison of the theoretical with 
the real discharges from an orifice, 317. 

Table of discharge of tubes of difi'erent en- 
largements, 322. 

Table of the comparison of discharge by pipes 
of different lengths, 323. 

Table of the comparison of discharge by ad- 
ditional tubes, 323. 

Table of the friction of fluids, 325. 

Table of discharges of a 6-inch pipe at seve- 
ral inclinations, 326. 

Table of the velocity of windmill sails, 333. 

Tabic of outside discharging turbine, 331. 

Table of inward discharging turbines, 332. 

Table of peculiar logarithms, 340. 

Table of useful logarithms, 345. 

Table of the specific gravity of various sub- 
stances, 394. 

Table of the weight of a foot in length of fiat 
and rolled iron, 395. 

Table of the weight of cast iron pipes, 395. 

Table of the weight of one foot in length of 
malleable iron, 396. 

Table of comparison, 396. 

Table of the weight of a square foot of sheet 
iron, 402. 

Table of the weight of a square foot of boiler 
plate from ^ of an inch to 1 inch thick, 403. 

Table of the weights of cast iron plates, 403. 

Table of the weight of mill-board, 405. 

Table of the weight of wrought iron bars. 406. 

Table of the proportional dimensions of nuts 
and bolts, 406. 

Table of the specific gravity of water at dif- 
ferent temperatures, 406. 

Table of the weight of cast iron balls, 407. 

Table of the weight of flat bar iron, 407. 

Table of the weight of square and round 
brass, 408. 

Table of taper T iron, 410. 

Table of sash iron, 410. 

Table of rails of equal top and bottom, 410. 

Table of temporary rails, 411. 

Table of natural sines, cosines, tangents, co- 
tangents, secants, and cosecants, to every 
degree of the qnadrant, 411. 

Table of inclined planes, showing the ascent 
or desc-^nt the yard, 430. 



Table of the weight of round steel, 403. 

Table of parallel angle iron of equal sides, 408. 

Table of parallel angle iron of unequal sides, 
409. 

Table of taper angle iron of equal sides, 409. 

Table of parallel T iron of unequal width and 
depth, 409. 

Table of change wheels for screw-cutting, 
435. 

Table of the diameters of wheels at their 
pitch circle, to contain a required number 
of teeth, 436. 

Table of the angle of windmill sails, 445. 

Table of the logarithms of the natural num- 
bers, from 1 to 100000, by the help of dif- 
ferences, 495 to 540. 

Table of log. sines, cosines, tangents, cotan- 
gents, secants and cosecants, for every de- 
gree and minute in the quadrant, 540 to 
576. 

Table of the strength of the teeth of cast iron 
wheels at a given velocity, 437. 

Table of approved proportions for wheels 
with flat arms, 441. 

Table showing the cover required on the 
steam side of the valve to cut the steam off 
at any part of the stroke, 228. 

Table showing the cover required, 227. 

Table showing the resistance opposed to 
the motion of carriages on difi'erent incli- 
nations of ascending or descending planes, 
429. 

Table showing the number of linear feet of 
scantling of various dimensions which are 
equal to a cubic foot, 95. 

Table showing the weight or pressure a beam 
of cast iron will sustain without destroying 
its elastic force, 292. 

Table showing the circumference of rope 
equal to a chain, 282. 

Table to correct parallel motion links, 248. 

Table of parallel T iron of equal depth and 
width, 410. 

Tables of cuttings and embankments, slopes, 
1 to 1 ; U to 1 : and 2 to 1, 97. 

Tables of the heights corresponding to differ- 
ent velocities, 389. 

Tables of the mechanical properties of the 
materials most commonly employed in the 
construction of machines and framings, 
280. 

Tangents, 360. 

Tangents and secants, to compute, 362. 

Taper angle iron, 410. 

Teeth of wheels in continuous circular motion, 
432. 

Teethofwheels, 422, 436. 

Temperature of steam, 172. 

Temperature and elastic force of steam, 188. 

Tension of chain-bridge, 414. 

Tetraedron, 89. 

Threshing machines, 445. 

Throttling the steam, 234. 

Timber measure, 93. 

Timber, to measm-e round, 95. 

Time, 7. 

Tonnage of ships, 461 to 494. 

Torsion, 279. 

Torsion and twisting, 286. 

Traction of carriages, 307. 

Transverse strength of bodies, 282. 



INDEX. 



591 



Transverse strain, 278. 

Transverse strain, time weight, 27.3. 

Trapezium, 47. 

Trapezoid, 47. 

Triangle, to find the area of, 46, 47. 

Trigonometry, 359. 

Trigonometry, spherical, 373. 

Troy weight, 7. 

Trussed beams, 291. 

Tubes, discharge of water through, 312. 

Tubular boilers, 257. 

Tui'bine water-wheels, 330. 

Ultimate pressure of expanded steam, 236. 

Undecagon, 47. 

Undershot wheels, 327, 443. 

Unguents, 299. 

Ungulas, cylindrical, 81. 

Ungulas, conical, 83, 84. 

Unit of length, 5. 

Unit of weight, 5. 

Unit of dry capacity, 5. 

Units of liquids, 5. 

Units of work, 269, 297, 414, 4i0. 

Universal pitch table, 442. 

Upper steam port, 229. 

Useful formula, 271. 

Use of the table of squares, cubes, &c., 127. 

VACuxJir, perfect one, 235. 
Vacuum below the piston, 251. 
Vacuo, bodies falling freely in, 388. 
Valves, different arrangements of, 233. 
Valve, length of stroke of, in inches, 228. 
Valve shaft, 147. 
Valve, safety, 224. 
Valve, slide, 225. 
Valve spindle, 171. 
Vapour in the cylinder, 229. 
Vein, contraction of fluid, 330. 
Velocity, force, and work done, 267. 
Velocity of steam rushing into a vacuum, 207. 
Velocity of smoke in chimneys, 209, 213. 
Velocity of piston of steam engine, 266. 
Velocity of threshing machines, millstones, 

boring, &c., 445. 
Velocity of wheels on ordinary roads, 307. 
Venturi, experiments of, on the discharge of 

fluids, 421. 
Versed sine, tabular, 52. 
Versed sine of parallel motion '-M:. 
Yersed sine, 359. 



1 Vertical sectional areas, 454. 
Virtual velocities, 424. 
Vis viva, principle of, calculations on, 276, 

388. 
Volume of a ship immersed, 456. 
Volume of steam in a cubic foot of water, 

202, 205. 

Water, modulus of elasticity of, 190. 
Water level, 214. 
Water, feed and condensing, 223. 
Water, spheroidal condition of, inboilers,2."6. 
"Water in boiler, and water level, 358. 
Water, discharge of, through different orifi- 
ces, 312, 318. 
Water wheels, 327. 

Water wheels, maximum velocity of, 443. 
Web of crank at paddle shaft centre, 136. 
Web of cross-head at middle, 139. 
Web of crank at pin centre, 142. 
Web at paddle centre, 252. 
Web of cross-head at journal, 140. 
Web of air-pump cross-head, 145. 
Wedge, 85. 

Wedge and screw, 430. 
Weights and measures, 5. 
Weights, values of, in decimal parts, 79. 
Weight, mass, gravity, 386 . 
Weirs, and rectangular apertures, 314, 323. 
Wheel and axle, 417. 
Wheel and pinion, 427. 
Wheels, drums, pulleys, 438. 
Windmills, 332. 
Wine measure, 8. 
Woods, 280. 

Woods, specific gravity, 394. 
Work done, weight, 267. 
Wrought iron bars, 406. 

Yard, 5. 

Yacht, admeasurement of, 466, 470. 

Yarns of ropes, 303. 

Yellow brass, 281. 

Yew, 280. 

Zixc, 280. 

Zinc, sheet, 288. 

Zone, spherical, 86. 

Zone, to find the area of a circular, 53. 

Zones of circles, to find the areas of, 64, 65, 



THE END. 



J 



W-- 



709 



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